UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Stability in the large of autonomous systems of two differential equations Mufti, Izhar-Ul Haq 1960

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1960_A1 M82 S8.pdf [ 3.64MB ]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0080607.json
JSON-LD: 1.0080607+ld.json
RDF/XML (Pretty): 1.0080607.xml
RDF/JSON: 1.0080607+rdf.json
Turtle: 1.0080607+rdf-turtle.txt
N-Triples: 1.0080607+rdf-ntriples.txt
Original Record: 1.0080607 +original-record.json
Full Text
1.0080607.txt
Citation
1.0080607.ris

Full Text

S T A B I L I T Y I N T H E L A R G E O F AUTONOMOUS S Y S T E M S O F TWO D I F F E R E N T I A L E Q U A T I O N S b y I Z H A R - U L HAQ M U F T I A T H E S I S S U B M I T T E D I N P A R T I A L FULFJLMEffl O F T H E E E Q U I E i M E N T S F O R T H E D E G R E E O F DOCTOR OF P H I L O S O P H Y i n t h e D e p a r t m e n t o f M A T H E M A T I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e s t a n d a r d r e q u i r e d f r o m c a n d i d a t e s f o r t h e d e g r e e o f D o c t o r o f P h i l o s o p h y . M e m b e r s o f t h e D e p a r t m e n t o f M a t h e m a t i c s T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l , I 9 6 0 . In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of Jy)A.lJLfcWvA.^/'X The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date G R A D U A T E STUDIES Field of Study: Applied Mathematics Functions of a Complex Variable R. R. Christian] Theory of Functions of a Real Variable R. A . Restrepo Differential Geometry D . Derry Linear Analysis and Group Representation H . F. Davis Other Studies: . Statistical Theory of Matter W. Opechowski Elementary Quantum Mechanics G . M . Volkoff Spectroscopy A . M . Crooker Quantum Theory of Radiation F. A . Kaempffer (Elu* Miiiuersiiy uf British OluUunbia Faculty of Graduate Studies PROGRAMME OF THE FINAL ORAL E X A M I N A T I O N FOR T H E D E G R E E O F DOCTOR OF PHILOSOPHY of IZHARUL HAQ MUFIT B.Sc. (Hons.) University of Sind, Pakistan 1951 M.Sc. University of Karachi, Pakistan 1953 IN ROOM 255, BUCHANAN BUILDING FRIDAY, APRIL 8, 1960 AT 3:00 P. M . COMMITTEE IN CHARGE D E A N G . M . S H R U M : Chairman E . L E I M A N 1 S W. S. T A Y L O R D . D E R R Y G . M . V O L K O F F M . M A R C U S R. E. B U R G E S S R. A . R E S T R E P O F. A . K A E M P F F E R External Examiner: PROF. J. P. L A S A L L E University of Notre Dame, Indiana S T A B I L I T Y IN T H E L A R G E OF A U T O N O M O U S S Y S T E M S OF T W O DIFFERENTIAL E Q U A T I O N S A B S T R A C T T h e object of this dissertation is to discuss the stability i n the large of the t r iv ia l solution for systems of two differential equa tions using qualitative methods (of course in combination with the construction of Lyapunov function). T h e right hand sides of these systems do not contain the t ime t explici t ly . First of a l l we discuss the system of'the type ^ = F ( * , y ) , =ip-), * = c x - dy, c j t o (1) These equations occur in automatic regulation. Using qualitative methods we determine sufficient conditions in order that the t r i v  i a l solution of system (1) be asymptotically stable i n the large. In this connection we note that a theorem proved by Ershov (Prikl. M a t . M e h . 18(1954), 381-383) is wrong. We then solve the problem of Aizerman for the systems of two equations, namely, for the systems | £ = f(x).+ ay., . dy_..=-bx.+ cy (2) and 4 ^ = a x + f ( y ) . . $ ! = b x + x y (3) dt v / / ' dt In the case of'system (2) we give^anew proof of a theorem which asserts that if c 2 +. ab i-.'o,. then the. t r ivial solution is asympto t ica l ly stable i n the large under the generalized Hurwitz conditions. T h e theorem was.first proved by Erugin. For system (3) M a l k i n showed that the trivial? solution is asymptotically stable in the large under the conditions: a + c < o, (acy - bf (y)) y ^ o for y jt'b and (4) I (acy - bf(y)) dy —> + oo as lyl-> + oo (5) We prove a similar theorem without the requirement of (5). We also discuss the stability in the large of the systems dx = ax + L (y) , dy. = f ( x ) + c dt 1 dt 2 W dx = f ( x ) = f (y). , dy. = b x + c y dt 1 c dt We consider again the system of the type (1) under assump tions as indicated by Ershov (Prikl. M a t . Meh. : 17(1953), 61-72) who has discussed various cases where the stability in the large holds. Not agreeing fully with the proofs of these theorems we give our own proofs. F inal ly we discuss the stability in the large of the systems dx = h « ( y ) x + ay , dy_ = h 2 ( x ) x + by dt 1 dt d i = xh„ (y) + ay , & = bx + h 9 M T - dt. 1 . dt 2 (6) under, suitable assumptions. A s a sample case we prove that if ab > o, then the t r iv ia l solution of system (6) is-asymptotically stable in the large under conditions: h (y) + h (x) < o, K< (y) h , (x) - ab ~? o for x jfc o,' y # o. . r. 2. -A B S T R A C T T h e o b j e c t o f t h i s d i s s e r t a t i o n i s t o d i s c u s s t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n f o r s y s t e m s o f t w o d i f f e r e n t i a l e q u a t i o n s u s i n g q u a l i t a t i v e m e t h o d s ( o f c o u r s e i n c o m b i n a t i o n w i t h t h e c o n s t r u c t i o n o f L y a p u n o v ' f u n c t i o n ) . T h e r i g h t h a n d s i d e s o f t h e s e s y s t e m s d o n o t c o n t a i n t h e t i m e t e x p l i c i t l y . F i r s t o f a l l we d i s c u s s ( S e c . 2.) t h e s y s t e m o f t h e t y p e = f W , ,. = cx - dy (1) T h e s e e q u a t i o n s o c c u r i n a u t o m a t i c r e g u l a t i o n . U s i n g q u a l i t a t i v e m e t h o d s we d e t e r m i n e s u f f i c i e n t c o n d i t i o n s i n o r d e r - t h a t t h e t r i v i a l s o l u t i o n o f s y s t e m ( l ) b e a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . I n t h i s c o n n e c t i o n we n o t e t h a t a t h e o r e m p r o v e d b y E r s h o v [7] i s w r o n g ( S e c . 2) . We t h e n s o l v e t h e p r o b l e m o f A i z e r m a n f o r t h e s y s t e m s o f t w o e q u a t i o n s ( S e c . 3), n a m e l y , f o r t h e s y s t e m s = f ( x ) + a y d x d t d t (2) b x + c y a n d 1 i i i g « ax + f ( y ) (3) If = b x + c y I n t h e c a s e o f s y s t e m (2) we g i v e a n e w p r o o f o f a 2 t h e o r e m w h i c h a s s e r t s t h a t i f c + a b * o , t h e n t h e t r i v i a l s o l u t i o n i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r t h e g e n e r a l i z e d H u r w i t z c o n d i t i o n s . T h e t h e o r e m w a s f i r s t p r o v e d b y E r u g i n [8], P o r s y s t e m (3) M a l k i n s h o w e d t h a t t h e t r i v i a l s o l u t i o n i s a s y m p  t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r t h e c o n d i t i o n s a + c < o , ( a c y - b f ( y ) ) y > o f o r y * o a n d y j ( a c y - b f ( y ) ) d y — + a s | y } _ » + ~ o We p r o v e a s i m i l a r t h e o r e m w i t h o u t t h e r e q u i r e m e n t o f yj ( a c y - b f ( y ) ) d y — * + « a s | y | - * + 00 o We a l s o d i s c u s s ( S e c . 4) t h e s t a b i l i t y i n t h e l a r g e o f t h e s y s t e m s § - . x ^ W , f - f 2(x) + c y d x = f ^ x ) + f 2 ( y ) , f £ = b x + c y We c o n s i d e r ( S e c . 5) a g a i n t h e s y s t e m o f t h e t y p e ( l ) b u t u n d e r a s s u m p t i o n s a s i n d i c a t e d b y E r s h o v w h o h a s d i s  c u s s e d v a r i o u s c a s e s w h e r e t h e a s y m p t o t i c s t a b i l i t y i n t h e l a r g e h o l d s . N o t a g r e e i n g f u l l y w i t h t h e p r o o f s o f t h e s e t h e o r e m s we g i v e o u r o w n p r o o f s . F i n a l l y we d i s c u s s ( S e c . 6 a n d 7) t h e . i v s t a b i l i t y in the large of the systems | | = ^ (y) x + ay , | j = h 2(x) x + by f | = xh/y) + ay > f t = b x + h2^ x^ y under suitable assumptions. A s a sample case we prove that i f ab > o,then the t r i v i a l solution of system (4) i s asymptotically stable i n the large under conditions h^y) + h 2(x) < o , h.jCy) h 2(x) - ab > o,for x * 0, y * o V TABLE OF CONTENTS Page Introduction 1 1. Some basic theorems on s t a b i l i t y 4 2. S t a b i l i t y i n the large of the system f - * * , y ) , g . (W 14 3. The problem of Axzerman for systems of two equations 25 4. A generalization of the problem of Aizer- man for two equations , 38 5. The sta b i l i t y i n the large of |f = P(x,y), f£ = f(«0 using quali tative methods only 42 6. The s t a b i l i t y in the large of H = ^ (y) x + ay, |f = ^ ( x ) x + b y 55 7. The st a b i l i t y in the large of |f = xh^y) + ay, |f = bx + h 2(x)y 67 8. Remarks 69 Bibliography .70 vi ACKNOWLEDGEMENTS The author wishes to acknowledge his indebtedness to Dr. E. Leimanis for suggesting the topic of this thesis, and for encouragement and advice received throughout the preparation of this thesis. He also wishes to express his thanks to the National Research Council of Canada whose financial assistance has made this study possible. 1 I N T R O D U C T I O N T h e i n v e s t i g a t i o n o f i n t e g r a l c u r v e s i n t h e l a r g e u s i n g g e o m e t r i c a l o r q u a l i t a t i v e m e t h o d s f o r a s y s t e m o f t w o d i f f e r e n t i a l e q u a t i o n s ( i . e . , o n a p l a n e ) w a s s t a r t e d b y P o i n c a r e ' , a n d c o n t i n u e d b y m a n y a u t h o r s d u r i n g t h e l a s t e i g h t y y e a r s . I n 1950 N . P . E r u g i n ([8],[9] ) , f o r m u l a t e d a g e n e r a l t h e o r e m o f q u a l i  t a t i v e n a t u r e f o r t h e s t a b i l i t y i n t h e l a r g e . We h a v e m a d e f r e  q u e n t u s e o f t h i s t h e o r e m i n o u r w o r k . T h e m a i n p u r p o s e o f t h e t h e s i s i s t o s t u d y t h e s t a b i l i t y i n t h e l a r g e o f s y s t e m s o f t w o d i f f e r e n t i a l e q u a t i o n s . T h i s p r o b  l e m i s s o l v e d s o m e t i m e s b y c o n s t r u c t i n g L y a p u n o v f u n c t i o n s , s o m e  t i m e s o n t h e b a s i s o f q u a l i t a t i v e m e t h o d s a n d s o m e t i m e s b y t h e c o m b i n a t i o n o f q u a l i t a t i v e m e t h o d s a n d t h e c o n s t r u c t i o n o f L y a p u n o v f u n c t i o n s . I t s h o u l d b e n o t e d t h a t i n s o l v i n g t h e p r o b  l e m o f s t a b i l i t y b y t h e s e m e t h o d s we d o n o t h a v e t o f i n d e i t h e r p a r t i c u l a r o r g e n e r a l s o l u t i o n s o f t h e d i f f e r e n t i a l e q u a t i o n . I n S e c t i o n 1, we r e v i e w t h e c o n c e p t s o f s t a b i l i t y i n t h e s e n s e o f L y a p u n o v a n d a s y m p t o t i c s t a b i l i t i e s i n t h e s m a l l a n d i n t h e l a r g e a n d g i v e c r i t e r i a f o r s t a b i l i t y i n t h e l a r g e b a s e d o n t h e c o n s t r u c t i o n o f L y a p u n o v f u n c t i o n s a n d o n q u a l i t a t i v e m e t h o d s . T o w a r d s t h e e n d o f t h e s e c t i o n w e c o n s t r u c t a L y a p u n o v f u n c t i o n f o r t h e e q u a t i o n o f t h e s e c o n d o r d e r ' 2 a n d g i v e s u f f i c i e n t c o n d i t i o n s w h i c h e n s u r e t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n o f t h e a b o v e e q u a t i o n . I n S e c t i o n 2, we d i s c u s s t h e s t a b i l i t y i n t h e l a r g e o f t h e f o l l o w i n g s y s t e m o f d i f f e r e n t i a l e q u a t i o n s u s i n g q u a l i t a t i v e m e t h o d s i n c o m b i n a t i o n w i t h t h e c o n s t r u c t i o n o f L y a p u n o v f u n c t i o n a n d o b t a i n c e r t a i n s u f f i c i e n t c o n d i t i o n s w h i c h g u a r a n t e e t h e s t a b i l i t y i n t h e l a r g e . I n S e c t i o n 3, we d i s c u s s t h e f a m o u s p r o b l e m o f A i z e r m a n f o r t h e s y s t e m s o f t w o e q u a t i o n s . I n S e c t i o n 4, a s o r t o f g e n e r a l i z a t i o n o f p r o b l e m o f A x z e r m a n i s d i s c u s s e d . I n S e c t i o n 5, we d i s c u s s a g a i n t h e s y s t e m o f e q u a t i o n s o f S e c t i o n 2 b u t u n d e r d i f f e r e n t a s s u m p t i o n s a n d e s t a b l i s h t h e s t a b i l i t y i n t h e l a r g e u s i n g q u a l i t a t i v e m e t h o d s o n l y . d x d t = P ( x , y ) = f (<y) , w h e r e = c x - d y , I n S e c t i o n s 6 a n d 7, we d i s c u s s t h e s t a b i l i t y i n t h e l a r g e o f d x d t h ( y ) x + a y h ( x ) x + a y d t a n d d x d t x h i ( y ) + a y b x + h u ( x ) y d t 3 mostly by qualitative methods. It may be noted that Gu,Cao-hao fl6] has considered a similar problem. He has discussed the s t a b i l i t y of | f = ^ ( y ) + , ( y ) I - - * 'W by constructing a Lyapunov function. 4 1. Some Basic Theorems On Stability. Let us consider a system of d i f ferential equations dx ± = X^ (x^, Xg, ...,x n,t) ( l . l ) of the perturbed motion. It is assumed that X i(o, o, t) = o ( i = 1, ...,n) and the right hand sides X^ of ( l . l ) are continuous functions with respect to a l l their arguments and satisfy the condition of unique ness of solutions of the system ( l . l ) in the region - «o £ X^ 4+oo - , t > O (1.2) If we denote the t o t a l i t i e s (x^, x 2, x n) and(X^, ..., X n) by x and X(x, t) respectively, each being (n x 1)matrix, then the sys tem ( l . l ) i s written i n the form x = X(x,t) (1.3) Since i t i s assumed that X(o,t) = o, equation (1.3) admits the t r i v i a l solution x(t) = o. The motion corresponding to this solution i s called unperturbed motion and motions corresponding to a l l other solutions are known as perturbed motions. Definition 1. The t r i v i a l solution x(t) = o i s called stable in the sense of Lyapunov i f , given a small f > o, there exists a % ( € >t Q) such that, f o r a l l perturbed motions x(t) for which x ( t Q ) | ^ % holds, the inequality |x(t)| <. € i s satisfied for t >y t >, o. o Definition 2. If the t r i v i a l solution i s stable in the above sense and every perturbed motion sufficiently close to i t i s such 5 t h a t I3-10. ix(t)| = o , t h e n we s a y t h a t t h e t r i v i a l s o l u t i o n x ( t ) = o i s a s y m p t o t i c a l l y s t a b l e i n t h e s m a l l o r i n t h e s e n s e  o f L y a p u n o v . D e f i n i t i o n 3« I f h o w e v e r | x ( t ) | — > o a s t —» oo. , n o m a t t e r w h a t t h e p o i n t ( X q , t ) m a y b e , t h e n t h e u n p e r t u r b e d m o t i o n i s s a i d t o b e a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . L e t u s d i s c u s s i n m o r e d e t a i l t h e i m p l i c a t i o n s o f t h e a b o v e d e f i n i t i o n s . B y s a y i n g t h a t t h e t r i v i a l s o l u t i o n i s s t a b l e i n t h e s e n s e o f L y a p u n o v , we u n d e r s t a n d t h a t a n y p e r t u r b e d m o t i o n s t a r t e d n e a r x = o p o s s e s s e s t w o p r o p e r t i e s ( i ) i t i s d e f i n e d f o r a l l t >/ t >/ o a n d ( i i ) s a t i s f i e s t h e i n e q u a l i t y [ x ( t ) | < £ f o r t h e same v a l u e s o f t a s i n ( i ) . T h e f i r s t p r o p e r t y i s n o t e x  p l i c i t l y s t a t e d i n t h e d e f i n i t i o n t h o u g h i t i s a l w a y s u n d e r s t o o d . E r u g i n f l l ] s h o w e d t h a t t h e b o u n d e d n e s s o f s o l u t i o n s i m p l i e s t h e e x i s t e n c e o f s o l u t i o n s f o r a l l t t o w h e n t h e r i g h t h a n d o ° s i d e s o f ( l . l ) a r e d e f i n e d a n d c o n t i n u o u s i n t h e r e g i o n (1.2). E x a m p l e s c a n b e g i v e n w h e r e t h e s o l u t i o n s a r e b o u n d e d e v e n i f t h e y a r e n o t d e f i n e d f o r a l l t >, t >/ o . T h i s c a n . h a p p e n , f o r e x a m p l e , i n t h e c a s e w h e r e t h e r i g h t h a n d s i d e s o f d i f f e r e n t i a l e q u a t i o n s ( l j ) a r e d e f i n e d f o r a l l x a n d t b u t a r e n o t b o u n d e d f o r a l l t > o ( E r u g i n ^12]). We n o w t u r n t o t h e d e f i n i t i o n o f a s y m p t o t i c s t a b i l i t y a c c o r d i n g t o L y a p u n o v . T h i s c o n c e p t i n c l u d e s i n i t s e l f t w o p r o  p e r t i e s o f t h e s o l u t i o n s o f s y s t e m ( l . l ) . One i s t h a t o f s t a b i l i t y i n t h e s e n s e o f L y a p u n o v a n d t h e o t h e r i s i^m|x(t)| = o . 6 T h e r e a r e c a s e s w h e r e t h e s o l u t i o n s o f t h e s y s t e m ( l . l ) p o s s e s s t h e s e c o n d p r o p e r t y b u t t h e t r i v i a l s o l u t i o n m a y n o t b e s t a b l e i n t h e s e n s e o f L y a p u n o v . O n e s u c h c l a s s o f a s y s t e m o f d i f f e r e n  t i a l e q u a t i o n s h a s b e e n g i v e n b y N . N . K r a s o v s k i J [ l 8 ] . O t h e r t y p e s o f s t a b i l i t i e s , e . g . u n i f o r m s t a b i l i t y , u n i f o r m a s y m p t o t i c s t a b i l i t y a r e a l s o f o u n d i n t h e l i t e r a t u r e . T h e s e t y p e s o f s t a b i l i t i e s h a v e b e e n c o n s i d e r e d b y J . L . M a s s e r a [23], I . G . M a l k i n [22], N . N . K r a s o v s k i i [20], a n d o t h e r s . F o r s o l v i n g t h e p r o b l e m s r e l a t i n g t o t h e s t a b i l i t y o f t h e t r i v i a l s o l u t i o n t h e m e t h o d s a r e d i v i d e d i n t o t w o g r o u p s . I n t h e f i r s t g r o u p we i n c l u d e a l l t h e m e t h o d s i n w h i c h e i t h e r p a r t i c u l a r o r g e n e r a l s o l u t i o n s o f t h e e q u a t i o n s o f t h e p e r t u r b e d m o t i o n a r e d e t e r m i n e d . I n t h e s e c o n d g r o u p t h e p r o b l e m o f s t a b i l i t y i s m a d e t o d e p e n d o n a f u n c t i o n V ( x , t ) s a t i s f y i n g c e r  t a i n p r o p e r t i e s . A s t h e a b o v e c l a s s i f i c a t i o n w a s d o n e b y L y a p u n o v we c a l l , t h e t w o m e t h o d s t k c L y a p u n o v ' * f i r s t a n d s e c o n d m e t h o d s . L e t V ( x , t ) d e n o t e a n y s c a l a r f u n c t i o n o f x , t , c o n t i n  u o u s a n d h a v i n g c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f t h e f i r s t o r d e r i n a d o m a i n . \x\ $ % , t >, o , w h e r e V ( o , t ) = o . D e f i n i t i o n 4. T h e f u n c t i o n V ( x , t ) i s c a l l e d s e m i d e f i n i t e i n a d o m a i n i f i t a s s u m e s v a l u e s o f t h e . s a m e s i g n i n t h a t d o m a i n ( t h e v a l u e z e r o i s a l s o a l l o w e d ) . D e f i n i t i o n 5. A f u n c t i o n W ( x ) i n d e p e n d e n t o f t i s s a i d t o b e p o s i t i v e d e f i n i t e i n a d o m a i n i f W ( x ) > 0 f o r a l l x * o a n d W(o) = o . 7 D e f i n i t i o n 6. We s h a l l s a y t h a t V ( x , t ) i s p o s i t i v e d e f i n i t e i n a d o m a i n , i f t h e r e e x i s t s a p o s i t i v e d e f i n i t e c o n t i n u o u s f u n c  t i o n W ( x ) s u c h t h a t V ( x , t ) >, W ( x ) i n t h e d o m a i n o f d e f i n i t i o n . D e f i n i t i o n 7 . We s h a l l s a y t h a t V a d m i t s o f a n i n f i n i t e l y s m a l l  u p p e r b o u n d i f g i v e n ( > o , t h e r e e x i s t s a % > o s u c h t h a t j V ( x , t ) | ^ €• f o r t >, o w h e n e v e r \x\ £ 8 • D e f i n i t i o n 8. I f f o r e v e r y M > o , t h e r e e x i s t s a n u m b e r N > o s u c h t h a t f o r J x ] N , t >, o f o l l o w s | v ( x , t ) | > M , t h e n V ( x , t ) i s s a i d t o b e i n f i n i t e l y l a r g e . D e f i n i t i o n 9. A d e f i n i t e f u n c t i o n V ( x , t ) , t h e t o t a l d e r i v a t i v e o f w h i c h w i t h r e s p e c t t o t i m e i n v i e w o f t h e p e r t u r b e d e q u a t i o n s i s e i t h e r - a s e m i d e f i n i t e f u n c t i o n o f a s i g n o p p o s i t e t o t h a t o f V ( x , t ) o r i s i d e n t i c a l l y e q u a l t o z e r o , i s c a l l e d a L y a p u n o v  f u n c t i o n . L y a p u n o v p r o v e d t h e f o l l o w i n g c l a s s i c a l r e s u l t o n a s y m p t o t i c s t a b i l i t y . T h e o r e m 1.1. I f f o r t h e d i f f e r e n t i a l e q u a t i o n s o f p e r t u r b e d  m o t i o n s t h e r e e x i s t s a L y a p u n o v f u n c t i o n , p o s s e s s i n g a d e f i n i t e  d e r i v a t i v e , a n d a d m i t t i n g o f a n i n f i n i t e l y s m a l l u p p e r b o u n d , t h e n  t h e u n p e r t u r b e d m o t i o n i s a s y m p t o t i c a l l y s t a b l e . I t m a y b e n o t e d t h a t t h i s t h e o r e m i s n o t r e v e r s i b l e . A s i m p l e e x a m p l e t o t h i s e f f e c t h a s b e e n g i v e n b y J . L . M a s s e r a [23}. T h e f o l l o w i n g t h e o r e m a n a s y m p t o t i c s t a b i l i t y i n t h e l a r g e c a n b e p r o v e d i n t h e s a m e w a y a s i s p r o v e d a t h e o r e m o n a s y m p t o t i c s t a b i l i t y i n t h e s e n s e o f L y a p u n o v b y J . L . M a s s e r a [23]. 8 T h e o r e m 1.2. I f t h e r e e x i s t s a n i n f i n i t e l y l a r g e p o s i t i v e d e f i   n i t e f u n c t i o n V ( x , t ) w h i c h p o s s e s s e s a n i n f i n i t e l y s m a l l u p p e r  b o u n d a n d w h i c h i s s u c h t h a t i t s . t o t a l t i m e d e r i v a t i v e i s n e g a   t i v e d e f i n i t e , t h e n t h e s o l u t i o n x = o i s a s y m p t o t i c a l l y s t a b l e  i n t h e l a r g e . T h e i n v e r s i o n o f t h i s t h e o r e m h a s n o t b e e n p r o v e d s o f a r i n i t s q u i t e g e n e r a l i t y . O n l y i n s o m e p a r t i c u l a r c a s e s t h i s h a s b e e n d o n e . 1.2. I n t h i s S e c t i o n w e c o n s i d e r t h e f o l l o w i n g s y s t e m o f d i f f e r e n t i a l e q u a t i o n s d x i d t ~ = X i ^ x 1 ' • • * » x n ^ ^ = 1 » ' " » n ) (1-2.1) w h e r e t h e r i g h t h a n d s i d e s a r e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s o f t h e v a r i a b l e s x , , . . . , x i n t h e r e g i o n - «*> < x . < + <*> , i = 1 , 2 , . . . , n . F u r t h e r m o r e X i ( o , o , . . . , o ) = o ( i = 1 , 2 , . . . , n ) . T h e t h e o r e m c o r r e s p o n d i n g t o T h e o r e m 1.1 i s t h e f o l l o w i n g T h e o r e m 1.2.1. I f t h e r e e x i s t s f o r t h e s y s t e m ( l . 2 . l ) a p o s i t i v e d e f i n i t e f u n c t i o n V ( x i , . . .TQ f o r w h i c h i s n e g a t i v e d e f i n i t e , t h e n t h e s o l u t i o n x = o i s a s y m p t o t i c a l l y s t a b l e i n t h e s e n s e o f  L y a p u n o v . T h i s t h e o r e m i s r e v e r s i b l e a n d we h a v e T h e o r e m 1.2.2. I f t h e t r i v i a l s o l u t i o n o f s y s t e m ( l . 2 . l ) i s a s y m p   t o t i c a l l y s t a b l e a c c o r d i n g t o L y a p u n o v , t h e n a p o s i t i v e d e f i n i t e V - f u n c t i o n e x i s t s s u c h t h a t 2j-g i s n e g a t i v e d e f i n i t e . 9 T h e a b o v e t w o t h e o r e m s s h o w t h a t t h e V - f u n c t i o n s c h a r a c  t e r i z e t h e a s y m p t o t i c s t a b i l i t y o f t h e z e r o s o l u t i o n i n t h e s e n s e o f L y a p u n o v . We g i v e a s i m p l e e x a m p l e t o s h o w h o ? i r T h e o r e m 1.2.1 i s a p p l i e d t o t h e p r o b l e m s c o n c e r n i n g t h e a s y m p t o t i c s t a b i l i t y i n t h e s m a l l . C o n s i d e r t h e s y s t e m d x 3 d t = y " X dy 3 dt = - X " y (1.2.2) t h e c h a r a c t e r i s t i c r o o t s o f t h e s y s t e m o f f i r s t a p p r o x i m a t i o n a r e i i . i . e . , t h e r e a l p a r t s o f t h e r o o t s a r e z e r o s . H e n c e L y a p u n o v * 4 f i r s t m e t h o d c a n n o t b e a p p l i e d . L e t u s t a k e t h e f o l l o w i n g a s V - f u n c t i o n V ( x , y ) = x 2 + y 2 C l e a r l y , t h i s f u n c t i o n i s p o s i t i v e d e f i n i t e . I t s t i m e d e r i v a t i v e b y v i r t u e o f (l.2.2) i s g i v e n b y d V o ( 1+ L at = ~ 2 ^ + r) w h i c h i s o b v i o u s l y n e g a t i v e d e f i n i t e . H e n c e t h e t r i v i a l s o l u t i o n x = o o f s y s t e m (l.2.2) i s a s y m p t o t i c a l l y s t a b l e i n t h e s m a l l . N o t e h e r e t h a t w e d i d n o t h a v e t o f i n d e i t h e r g e n e r a l o r p a r t i c u  l a r s o l u t i o n s o f t h e s y s t e m i n o r d e r t o d e c i d e t h e s t a b i l i t y p r o b  l e m . We n o w s h o w b y a n e x a m p l e t h a t t h e V - f u n c t i o n s w h i c h g u a r a n t e e t h e a s y m p t o t i c s t a b i l i t y i n t h e s m a l l a r e n o t g o o d e n o u g h f o r t h e e s t a b l i s h m e n t o f t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n . C o n s i d e r t h e s y s t e m o f t w o e q u a t i o n s 10" If * y - *(*) . at = - • ( * ) ( 1 * 2 - 3 ) where * (o) = o, x * (x) > o for x * o. Following Malkin [2l] the V-function for this system can be taken 2 x as V(x,y) = -gy + J *(x) dx o Its total time derivative in view of (1.2.3) i s given by f - -<*«>2 Clearly, V(x,y) i s positive definite and < o for x $ o and ilZ = o for x = o. It can be shown that i n this case asymptotic dt dV s t a b i l i t y i n the small holds even i f -gfc i s not negative definite, x Now i f J * (x) dx —f-* + 0 0 as |xl—» + 0 0 , then i t i s possible o (Pliss [25]) that the stability in the large may not hold,i.e., we can show that there exist trajectories going to i n f i n i t y for t —» + 00 The above example shows that i t becomes necessary to put an extra condition on the V-functions i n order to realize asymp tot i c s t a b i l i t y in the large. The V-function should be such that V(x^ , X 2 , ••• •;. > x n) ^ C, C > o, defines a bounded region contain ing the origin for a l l C. Because then we can be sure of the solu tions being bounded and defined for a l l t t >, o, no matter o what the i n i t i a l point may be. Our purpose is served i f we im pose on the V-function an additional requirement of being i n f i n i t e  l y large, since i t i s known (Erugin (14] ) that V(x) possessing the property V(x) > o for x * o and V(o) = o does not define the 11 r e g i o n V ( x ) $ C , C > o w h i c h i s a l w a y s b o u n d e d . T h e f o l l o w i n g t h e o r e m s o n s t a b i l i t y i n t h e l a r g e a r e d u e t o E . A . B a r b a s h i n a n d N . N . K r a s o v s k i i [2]. T h e o r e m 1.2.3. I f t h e r e e x i s t s a n i n f i n i t e l y l a r g e p o s i t i v e d e f - d V i n i t e f u n c t i o n V ( x ) , t h e t o t a l t i m e d e r i v a t i v e -g^- o f w h i c h b y v i r t u e o f t h e p e r t u r b e d e q u a t i o n s i s n e g a t i v e d e f i n i t e , t h e n t h e  t r i v i a l s o l u t i o n o f ( l . 2 . l ) i s a s y m p t o t i c a l l y s t a b l e f o r a r b i   t r a r y i n i t i a l d i s t u r b a n c e s . T h e o r e m 1.2.4. I f t h e t r i v i a l s o l u t i o n x ( t ) = o i s a s y m p t o t i c a l l y  s t a b l e i n t h e l a r g e , t h e n t h e r e e x i s t s a c o n t i n u o u s l y d i f f e r e n - t i a b l e i n f i n i t e l y l a r g e p o s i t i v e d e f i n i t e f u n c t i o n V " ( x ) h a v i n g  n e g a t i v e d e f i n i t e d e r i v a t i v e w i t h r e s p e c t t o t i m e p r o v i d e d t h a t  a l l s o l u t i o n s c a n b e c o n t i n u e d t o t h e i n t e r v a l - ° ° < t 4 o . I t w a s p o i n t e d o u t b y E r u g i n [10], i n v e r t i n g a t h e o r e m o f W i n t n e r [30], t h a t n o t e v e r y s y s t e m o f t y p e ( l . 2 . l ) p o s s e s s e s t h e p r o p e r t y t h a t a l l i t s s o l u t i o n s b e c o n t i n u a b l e o n t h e w h o l e i n t e r v a l - « * < t < + ° ° . I t h a s b e e n s h o w n ([4], [31!) t h a t t h e r e q u i r e m e n t o f c o n t i n u a t i o n o f s o l u t i o n s o n t h e n e g a t i v e t - a x i s i n T h e o r e m 1.2.4 i s n o t e s s e n t i a l . A n i m m e d i a t e g e n e r a l i z a t i o n o f T h e o r e m 1.2.3 i s t h e f o l l o w i n g , T h e o r e m 1.2.5. L e t t h e r e e x i s t a n i n f i n i t e l y l a r g e p o s i t i v e d e f   i n i t e f u n c t i o n V ( x ) a n d a s e t M s u c h t h a t 4 r < © o u t s i d e M-iSL £ o o n M d t ' d t 12 Let.the set M possess the property that an arbitrary intersection  of the sets V = C (g * o) and M does not contain the positive  half trajectory of the system (l.2.l) then the t r i v i a l solution  x = o of system (1.2.l) i s asymptotically stable for arbitrary  i n i t i a l disturbances. As an example of Theorem 1.2.5 we consider the differen t i a l equation + * (dx) g(x) + f (x) (§&) = o . (1.2.4) dt^ cit dt This can be thrown into the form dt * § = -•(y) g(x) - f(x) *(y) (1.2.5) We as sume that x f(x) > o for x +- o, f(o) = o; g(x) > o, (1.2.6) <V (y) > o and y • (y) > o for y $ o,4>(o) = o Furthermore,it i s assumed that the right hand sides of (1.2.5) satisfy the conditions guaranteeing the existence and uniqueness of solutions of (1.2.5). We construct the following Lyapunov function for the system (1.2.5) x y V(x,y) = | f(x) dx + j - ^ y ) dy o o Clearly,V(x,y) is positive definite. Let us compute i t s total time derivative in view of equations (1.2.5). 1 3 § = f(x) y + ^ } [ - <P (y) g(x) - f ( x ) * ( y ) ] = " y s ^ < ° f o r y * ° 3 ° for y = o It is easy to see that y = o does not contain a posi tive half trajectory of the system ( 1 . 2 . 5 ) except the origin. If we now assume that x y f f(x) dx — » * for |x| -» «*» ; j ; j j j r ^ dy-*-*>for | V \ - * *> o o 7 ( 1 . 2 . 7 ) then V(x,y) i s i n f i n i t e l y large. Thus we prove the following Theorem 1 . 2 . 6 . If the conditions ( 1 . 2 . 6 ) and ( 1 . 2 . 7 ) are satis  f i e d then the t r i v i a l solution of Q - . 2 . 4 ) i s asymptotically stable  i n the large. It may be remarked here that the construction of su i t  able Lyapunov functions is possible in a very small number of ex amples (see [ 3 ] , [ 5 ] , [ 1 9 ] , [ 2 4 ] , [ 2 7 ] , and [ 2 9 ] ) . In 1 9 5 0 Erugin [ 8 ] proved the following theorem for the system of two equations ,i.e., for § = P(x,y) ( 1 . 2 . 8 ) Theorem 1 . 2 . 7 . (Erugin) We assume that (i) the point ( 0 , 0 ) i s the only point of equilibrium, ( i i ) the unperturbed motion x = o = y i s asymptotically stable and consequently any motion started in a certain region ( <: ) 14 x 2 + y 2 * * (1.2.9) possesswthe -property x(t) —» o, y(t)~-» o as t -» *> , (1.2.10) ( i i i ) a straight line L(o,°») going, to i n f i n i t y from the point  (o,o) i s intersected by the motions i n one direction  only for t —»«•», (iv) the motions having bounded polar angles are bounded, (v) there are no periodic motions; then a l l the motions possess the property (l.2.10). The above theorem has been generalized to the case of a system of n equations by V.A. P l i s s [26], 2. Stability in the large of the system ^  = F(x,y), §fe = f ( «* ). In this section we shall consider the system of equations i - ^ (2.1) |f = f ( * ' ) , ^ = cx - dy where c and d are constants, c * o ; the functions F(x,y), f ( ) are continuous and F(o,o) = o, f(o) = o. Besides, the fulfilment of conditions of uniqueness of the solution x = o = y i s assumed. The above system was considered i n the works of Ershov ([6], [7]) and Krasovskii [19]. Following Krasovskil we transform the system (2.l) to the following form (2.2) by the change of de pendent variables expressed by the relations «•* = cx - dy 15 The above transformation i s non-singular because c is not assumed equal to zero. We then have If -'• (2.2) where * («*,y) = c F ( r + d y , y) - d t(f) . c Krasovskii constructed the following Lyapunov . func tion for the system (2.2) •* y V(«-,y) = j f ( « - ) d r - j*(o,y) dy o o and using Theorem 1.2.5 proved the following theorem.* Theorem 2.1. If the conditions c f(«*) 7 o for * o (2.3) y+(o,y)< o f o r y * o (2.4) «* [•(«* ,y) - * (o,y) ] <• o f o r «* * o (2.5) and J f ( r ) d r = ~ , j+(o,y) dy = oo (2.6) are satisfied then the t r i v i a l solution x = o = y of system (2.1)  i s asymptotically stable in the large. It may be remarked that conditions (2.3) and (2.4) can be replaced by the following conditions *. * f(<r) * o f o r «* * o (2.3) y *(o,y) > o for y * o (2.4) Ershov [7] claimed that Theorem 2.1 holds without the requirement of conditions (2.6). In fact,he stated the following 16 theorem: Theorem 2.2. If conditions (2.3), (2.4) and (2.5) are satisfied  f o r the system (2.l), then the t r i v i a l solution of system (2.l)  i s asymptotically stable i n the large. The following example shows that conditions (2.6) can not be removed in general. Example. Consider the system of equations (2.7) where f(x) i s defined as below ,-2x f(x) = T 2 — f o r x >, 1 \ / -| + e~x ~-2 = — £ — . % for x < 1 1 + e""1 Obviously, xf(x) > o for x * o and f(o) = o y *(°,y) = y(-y -f(°)) = - y < o for y * o • " [ * ( r,y) - *(o,y)] = x(-y -f(x) + y) = -x f(x) C o f o r x * o Moreover,it i s not d i f f i c u l t to show that f(x) is continuous and satisfies the Lipschitz condition. Thus a l l the conditions of Theorem 2.2 are satisfied. We show that the t r i v i a l solution of this system i s asymptotically stable in the sense of Lyapunov but not i n the large. The s t a b i l i t y in the small follows from ttee f o l  lowing Lyapunov function x ? V(x,y) = J" f(x) dx + i y \ o 17 We now show that there exist trajectories going to i n f i n i t y f o r t —* + 0 0 . . It i s easy to verify that y = -e is a particular inte gral of the system on the interval 1 £ x < 0 0 passing through —1 —3 the point (1, -e ) at t = o. We show that along the curve y = -e > o.i.e. x increases with the increase of time, dt ' cix / \ —x —2x ~x dt = T " F ( X ) = + E ' TV^ -x = > o for x >,1 We integrate |x = e ~ x along the trajectory y = -e" x and have dt 1+e~x x t x t J 1 + e" X dx = J dt or e x + x | = t | 1 e _ X o 1 o or x + e x - e - 1 = t Prom the last equation i t follows that as t —• + <x> , x —-» + 0° , i.e.^the positive half trajectory y = -e~ x of the system (2.7) tends to i n f i n i t y as t —-* + »» . Hence i t follows that the t r i v i a l solution i s not asymptotically stable i n the large. 2.2. Let us consider the system dt •ff = f ( r ) , = cx -dy, c * o (2.2.1) dt under the conditions: F(o,o) = o, f(o) = o (2.2.2) > o f o r «* * o (2.2.3) «* [•(«#,y) - *(o,y)] <• o for «- * o (2.2.4) 1 8 y * ( o , y ) < o for y * o ( 2 . 2 . 5 ) w h e r e 4> (c,y) = c F(JLL&L, y ) - d f ( •• ) ( 2 . 2 . 6 ) c B e s i d e s , t h e f u l f i l m e n t o f c o n d i t i o n s o f u n i q u e n e s s o f t h e s o l u t i o n x = o = y i s a s s u m e d . A s b e f o r e we r e d u c e t h e s y s t e m ( 2 . 2 . 1 ) t o t h e f o l l o w i n g s y s t e m jf - * (2.2.7) We c o n s i d e r t h e p o s i t i o n s o f t h e c u r v e s r e p r e s e n t e d b y t h e r i g h t - h a n d s i d e s o f ( 2 . 2 . 7 ) o n t h e ( « * , y ) p l a n e , i . e . , o f <P (a^y) = o a n d f(o*) = o . S i n c e «• f ( « * ) > o f o r <r* + o a n d f ( o ) = o , f ( « * ) = 0 o n l y w h e n * = o , i . e # J f ( «• ) = o r e p r e s e n t s t h e y - a x i s . We n o w t u r n t o t h e c u r v e r e p r e s e n t e d b y 4» ( « * , y ) = o . We o b s e r v e t h a t 4» (0,0) = o a n d o n t h e «* - a x i s 4> ( « * , y ) > o f o r < o , <*»(«f,y) < o f o r «* > o ; o n t h e y - a x i s * ( * " , y ) > o f o r y < o , * (o*,y) 0 f ° r y > o . F r o m t h e s e f a c t s i t f o l l o w s t h a t 4> («*,y)<o_, ff* 7 o , y 7 o a n d <t» ( * * , y ) > o f o r «* < o , y < o . I n d e r i v i n g t h e s e c o n c l u s i o n s we h a v e m a d e u s e o f t h e c o n d i t i o n s ( 2 . 2 . 4 ) a n d ( 2 . 2 . 5 ) . T h u s i t f o l l o w s t h a t «fr ( « * , y ) c h a n g e s s i g n i n t h e s e c o n d a n d f o u r t h q u a d r a n t s a n d h e n c e t h e c u r v e 4> ( » * , y ) = o l i e s i n t h e s e c o n d a n d f o u r t h q u a d r a n t s . I t i s e a s y t o s e e t h a t d «• = <p ( « * , y ) > o f o r t h e p o i n t s l y i n g t o t h e l e f t d t o f t h e c u r v e <P ( » * , y ) = o d «* . -rr— = 4 » ( * ' , y ) < o . f o r t h e p o i n t s l y i n g t o t h e r i g h t o f t h e c u r v e «> (r - , y ) = o 19 •^2. = f ( «*) > o f o r t h e p o i n t s l y i n g t o t h e r i g h t o f y ^ a x i s = f ( • * ) < , o f o r t h e p o i n t s l y i n g t o t h e l e f t o f y - a x i s T h e c u r v e * ( « * , y ) = o a n d t h e c o - o r d i n a t e a x e s d e c o m  p o s e t h e p l a n e ( < f , y ) i n t o s i x r e g i o n s . y ( t ) i s m a x i m u m o n t h e y - a x i s f o r y > o a n d m i n i m u m f o r y < o ; * ( t ) i s m a x i m u m o n t h e c u r v e 4> ( « * , y ) = o f o r o* > o a n d m i n i m u m f o r «* <. o . T h e d i r e c  t i o n o f m o t i o n i s i n d i c a t e d i n f i g . 1. We i n t r o d u c e p o l a r c o - o r d i n a t e «* = r c o s 8 , y = r s i n $ , t h e n r = <s* c o s 6 + y s i n 8 a n d r 8 = - *» s i n s + y c o s T h e s i g n s o f r a n d 6 i n d i f f e r e n t r e g i o n s a r e g i v e n a s b e l o w : (1,4) r m a y b e >^ o , 6 > o (2,5) f m a y b e \ o , Q > - o (3,6) r < o , d m a y b e ^ o T h e L y a p u n o v . f u n c t i o n f o r t h e s y s t e m (2.2.7)is f yr V ( c , y ) = J f ( c ) d r - J * ( o , y ) d y o o I t s t o t a l t i m e d e r i v a t i v e i n v i e w o f e q u a t i o n s (2.2.7) i s V = f(«0 ^ • ( • • , y ) - * ( o , y ) ] < . o f o r * o = o f o r •* * o O b v i o u s l y , V ^ y ) i s p o s i t i v e d e f i n i t e a n d V i s o f n e g a t i v e s i g n . N o w i L f l = < * ( o , y ) f o r «* = o a n d i t i s d i f f e r e n t f r o m z e r o u n l e s s y = d t 20 T h i s m e a n s t h a t <?* = o d o e s n o t c o n t a i n a n y o t h e r p o s i t i v e h a l f t r a j e c t o r y o f t h e s y s t e m (2.2.7) t h a n t h e o r i g i n . W h e n c e f o l l o w s t h e a s y m p t o t i c s t a b i l i t y i n t h e s e n s e o f L y a p u n o v o f t h e t r i v i a l s o l u t i o n o f s y s t e m (2.2.7). T h u s we h a v e s h o w n ( i ) t h e p o i n t ( 0 , 0 ) i s t h e o n l y p o i n t o f e q u i l i b r i u m , ( i i ) t h e u n p e r t u r b e d m o t i o n i s a s y m p t o t i c a l l y s t a b l e a c c o r d i n g t o L y a p u n o v , ( i i i ) t h e r e a r e n o p e r i o d i c m o t i o n s , s i n c e f o r (2.2.7) i s c o n  s t r u c t e d a L y a p u n o v f u n c t i o n . S i n c e t h e «• p o s i t i v e h a l f a x i s i s i n t e r s e c t e d b y t h e m o t i o n s i n o n e d i r e c t i o n o n l y , i t c a n b e t a k e n f o r t h e s t r a i g h t l i n e L ( o , o o ) a p p e a r i n g i n T h e o r e m 1.2.7. T h u s a l l t h e c o n d i t i o n s o f T h e o r e m 1.2.7 a r e s a t i s f i e d e x c e p t t h e f o u r t h ^ i . e . , t h e m o t i o n s h a v i n g b o u n d e d p o l a r a n g l e s a r e b o u n d e d . We n o w i n d i c a t e w h a t a d d i t i o n a l c o n d i t i o n s a r e t o b e i m p o s e d i n o r d e r t o r e a l i z e t h e f o u r t h c o n  d i t i o n o f T h e o r e m 1.2.7. F I G . i 21 L e t a m o t i o n M ( t ) s t a r t i n a r e g i o n , , s a y (6). A n y m o t i o n M ( t ) s t a r t e d i n t h e r e g i o n (6) o r e n t e r i n g t h i s r e g i o n e i t h e r t e n d s t o t h e o r i g i n o r g o e s o u t o f t h i s r e g i o n a n d e n t e r s t h e r e g i o n ( l ) . T h i s f o l l o w s f r o m t h e f a c t t h a t f < o i n t h i s r e g i o n . A f t e r e n  t e r i n g t h e r e g i o n ( l ) t h e m o t i o n M ( t ) e i t h e r c r o s s e s t h e y - a x i s o r t e n d s t o i n f i n i t y a l o n g t h e y - a x i s , b u t i t c a n n o t g o t o t h e o r i g i n s i n c e Q > o . We a r e t h u s l e d t o i m p o s e t h e f o l l o w i n g C o n d i t i o n A ' . We a s s u m e t h a t * ( o % y ) s a t i s f i e s s u c h c o n d i t i o n s c a l l e d A t h a t t h e m o t i o n e n t e r i n g o r s t a r t i n g i n t h e r e g i o n s ( l ) a n d (4) l e a v e s t h e s e r e g i o n s w i t h t h e i n c r e a s e o f t i m e . T h i s c o n d i t i o n g u a r a n t e e s t h a t t h e r e a r e n o m o t i o n s w i t h b o u n d e d p o l a r a n g l e s i n t h e r e g i o n s ( l ) a n d (4). We i m p o s e a n o t h e r c o n d i t i o n c a l l e d B . C o n d i t i o n B . We a s s u m e t h a t * (•*, c ) = o h a s a s o l u t i o n f o r a l l c . I f C o n d i t i o n A i s s a t i s f i e d t h e n t h e m o t i o n M ( t ) e n t e r s t h e r e g i o n (2). We n o w s h o w t h a t t h e m o t i o n M ( t ) l e a v e s t h e r e g i o n (2) w i t h t h e i n c r e a s e o f t i m e i f C o n d i t i o n B i s s a t i s f i e d . I n f a c t i f * ( « ' , c ) = o h a s a s o l u t i o n f o r a l l ct±.e ,t±f y = c i n t e r s e c t s t h e c u r v e • ( « * , y ) = 0, t h e m o t i o n a f t e r e n t e r i n g t h e r e g i o n (2) c a n n e i t h e r c r o s s t h e l i n e y = c,since ^ < o i n t h i s r e g i o n n o r i t c a n e n t e r t h e o r i g i n b e c a u s e 6 > o i n r e g i o n (2). T h e r e f o r e t h e m o t i o n m u s t i n t e r s e c t t h e c u r v e • (<*%y) = 0. S i m i l a r r e a s o n  i n g c a n b e c a r r i e d o u t i n t h e r e g i o n s (3), (4) a n d ( 5 ) . T h e a b o v e a n a l y s i s s h o w s t h a t t h e r e a r e n o m o t i o n s w i t h , b o u n d e d p o l a r a n g l e s 22 i n t h e r e g i o n s ( l ) , (2), (4) a n d (5) a n d t h a t t h e m o t i o n s w i t h b o u n d e d p o l a r a n g l e s c a n o c c u r o n l y i n t h e r e g i o n s (3) a n d (6) a n d a s p r o v e d a b o v e t h e y a r e b o u n d e d . T h u s a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d . T h e C o n d i t i o n B w a s i m p o s e d w i t h a v i e w t o e n s u r e t h a t i n t h e r e g i o n s (2) a n d (5) t h e r e a r e n o m o t i o n s w i t h b o u n d e d p o l a r a n g l e s . T h i s c a n a l s o b e a c h i e v e d i f we a s s u m e t h a t i n t h e s e r e g i o n s 8 > 6 > o . We c a l l t h i s c o n d i t i o n a s C o n d i t i o n C . We n o w c o l l e c t t h e s e r e s u l t s i n t h e f o l l o w i n g : T h e o r e m 2.2.1. I f C o n d i t i o n s (2.2.2) (2.2.5) a r e s a t i s f i e d a n d i f e i t h e r C o n d i t i o n s A a n d B o r C o n d i t i o n s A a n d C a r e s a t i s f i e d ,  t h e n t h e t r i v i a l s o l u t i o n o f (2.2 . l ) i s a s y m p t o t i c a l l y s t a b l e i n  t h e l a r g e . I t m a y b e r e m a r k e d t h a t C o n d i t i o n s B a n d C a r e i n d e p e n  d e n t , i . e . , i f C o n d i t i o n B h o l d s t h e n C o n d i t i o n C m a y o r may n o t h o l d o r v i c e v e r s a . I n t h e l i n e a r c a s e , i . e . , w h e n t h e r i g h t h a n d s i d e s o f s y s t e m (2.2.1) a r e l i n e a r , h o w e v e r , b o t h h o l d . L e t u s c o n s i d e r a f e w e x a m p l e s n o w . E x a m p l e 1. C o n s i d e r t h e s y s t e m d x - x -1 — = - y + e - 1 d t J (2.2.8) = x d t X O b v i o u s l y , t h i s s y s t e m s a t i s f i e s t h e c o n d i t i o n s (2.2.2)— (2.2.5). We v e r i f y t h a t i n t h i s c a s e C o n d i t i o n A i s s a t i s f i e d . L e t u s f o r t h e s a k e o f d e f i n i t e n e s s a s s u m e t h a t t h e m o t i o n M ( t ) 23 i s i n t h e r e g i o n ( l ) . We a s s u m e t h a t t h e C o n d i t i o n A i s n o t s a t i s - a f i e d . T h e n t h e m o t i o n M ( t ) g o e s t o i n f i n i t y a s t — * •* a n d d u r i n g t h i s y ( t ) — » — . F r o m t h e f i r s t e q u a t i o n o f (2.2.8) f o l l o w s t h a t - ^ b e c o m e s i n - , f i n i t e l y l a r g e b u t n e g a t i v e w h i c h m e a n s t h a t M ( t ) c a n n o t r e m a i n i n t h e f i r s t q u a d r a n t a n d h e n c e o u r a s s u m p t i o n 3' i s n o t t r u e . I t i s n o t d i f f i c u l t t o _ . _ 0 FIG, 2 v e r i f y t h a t t h e s t r a i g h t l i n e y = c d o e s n o t i n t e r s e c t t h e c u r v e - y + e x - 1 = o f o r a l l c ( s e e f i g . 2) H e n c e C o n d i t i o n B i s n o t s a t i s f i e d . H o w e v e r C o n d i t i o n C h o l d s . I n f a c t r e = - x s i n e + y c o s 8 = - s i n a ( - y + e x-1 ) + x c o s 9 a n d 6 = - 3 i n 9 ( - y + e x-1 ) + c o s 2 8 > € > 0 i n t h e r e g i o n s (2) a n d (5). T h i s e s t a b l i s h e s t h e a s y m p t o t i c s t a  b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n o f (2.2.8). E x a m p l e 2. C o n s i d e r t h e e q u a t i o n f £ + • Cf£) g(y) + f ( y ) = o (2.2.9) T h i s e q u a t i o n c a n b e t h r o w n i n t o t h e s y s t e m § = - *U) g ( y ) - f ( y ) d t " b y w r i t i n g x f o r dy_ d t We a s s u m e t h a t t h e f o l l o w i n g c o n d i t i o n s a r e f u l f i l l e d : 24 y t(y) > o, f(o) = o and either h^(x) > o for x * o and g(y) > o for a l l y or h^ (x) < o for x * o and g(y) < o for a l l y where h.(x) = * ( x ) for x * o 1 x The Lyapunov function for this system is X y V(x,y) = J xdx + J f(y)dy o o Then V = x(-*(x) g(y) - f(y)) + f(y)x = - x *(x) g(y) It is not d i f f i c u l t to see that in this case Condition C of Theorem 2.2.1 i s satisfied. Thus i f Condition A i s also sat i s f i e d then the t r i v i a l solution of (2.2.9) i s asymptotically stable i n the large. Example 3. Consider the system | f = 4>(x) + £ ( y ) (2.2.10) = f (x) dt 1 w •^ or the system (2.2.10) the V function i s x y V(x,y) = J f ^ ) dx - J f 2 ( y ) dy o o Tnen V(x,y) = f 1(x)*(x) We subject (2.2.10) to the following conditions: <•> (o) = o , h3(x) < o for x £ o , where h,(x) = ti$)for x£ o J x (2.2.11) f1 (o) = o,xfj (x) > o f o r x * o , f 2 (o) = o and y f 2 ( y ) < o for y * o 25 The t r i v i a l solution of (2.2.10) is thus asymptotically stable i n the large i f i n addition to (2.2.1l) the Conditions A and B or A and C are satisfied. Example 4. Consider the equation of the second order ^ + . g < y ) - o (2.2.12) We write i t i n the formof jbsystem of equations |f = - f (x ,y) x - g(y) dt X We assume that g(o) = o, y g(y) > o for y * o and f(x»y) > ° f ° r x + o It i s easy to verify that asymptotic stability in the large holds i f Condition A i s satisfied. 3. The Problem of Aizerman. In 1949 Aizerman [ l ^ proposed the following problem. Let there be given a system of linear differential equa tions d x1 y dxi t — dt" = h 1 a i J x J + > dt~ = f r i a i j X J (3-D ( i = 2,3,-.-fn) Suppose that for the given constants a^^ ( i = 1 , •;«..«->n,« j=1 ,».»jn) and for an arbitrary value of 'a' from the interval J. < a <. P a l l the roots of the characteristic equation of (3.l) have negative 26 r e a l p a r t s . L e t ax^ . b e r e p l a c e d b y fCx^) i n ( 3 « l ) . We t h e n h a v e dx1 n dt" " j ~ (3.2) = r = 1 1^ j X J + f ( x k ) d x ± " " d t = j = 1 «U X j ( i = 2 " - - n ) I t i s r e q u i r e d t o f i n d o u t w h e t h e r t h e t r i v i a l s o l u t i o n x^ = X g = • = x n = o o f s y s t e m (3.2) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e o r n o t , f o r « a r b i t r a r y c h o i c e o f ^ c o n t i n u o u s f u n c t i o n A f ( x ^ . ) , w h i c h r e d u c e s t o z e r o f o r x ^ = o a n d w h i c h s a t i s f i e s t h e i n e q u a l i t y L x l < ^ f ( x k ) < P x 2 . f o r ^ * o (3.3) T h e a n s w e r t o t h e a b o v e p r o b l e m i s i n t h e n e g a t i v e ( P l i s s f 2 5 j ) . T h e i n t e r e s t i n t h e p r o b l e m i s r e v i v e d i f we s l i g h t l y c h a n g e t h e a b o v e p r o b l e m a n d a s k o u r s e l v e s t h e f o l l o w i n g q u e s t i o n s . P o r w h a t v a l u e s o f a . . t h e a n s w e r t o t h e a b o v e p r o b l e m i s i n t h e a f f i r m a  n t ) t i v e a n d f o r w h a t v a l u e s i n t h e n e g a t i v e . I f t h e t r i v i a l s o l u t i o n i s n o t a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r t h e g e n e r a l i z e d H u r w i t z 1 c o n d i t i o n s ( i . e . , t h e c o n d i t i o n (3.3) a n d f ( o ) = o ) , w h a t a d d i t i o n a l a s s u m p t i o n s s h o u l d b e m a d e o n ^(Xjj.) s o t h a t t h e t r i v i a l s o l u t i o n b e c o m e s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . We s h a l l d i s  c u s s h e r e t h e c a s e w h e n n = 2 . T h e c a s e n = 3, k = 2 h a s b e e n c o m  p l e t e l y s o l v e d b y P l i s s [28]. F i r s t o f a l l we t a k e n = 2 = k , i . e . , we c o n s i d e r t h e s y s t e m f = a x + , f ( y ) (3.4) d t = b x + °y 27 under the assumptions that f(o) = o , a + c < o , a c - bh(y) > o^y * o (3.5) where h(y) = f (y) for y * o y Besides, the uniqueness of the t r i v i a l solution i s assumed. System (3.4) was considered by N.P. Erugin [9} and I.G. Malkin[2l] . Malkin [2l] showed that the t r i v i a l solution of system (3.4) is asymptotically stable in the large i f for sufficiently large values of Jy| the inequality ac - b h(y) > 6 holds. This condition can be relaxed to the condition that y J (ac - bh(y) y dy — » +°° as | y\ —» + «* o (3.6) We show that asymptotic stability i n the large of the t r i v i a l solution of (3.4) holds without the requirement of con dition (3.6). Por b = o, from (3.5) follows that a < o, c < o and an immediate integration of the system (3.4) shows that the equilibrium i s asymptotically stable for arbitrary i n i t i a l distur bances and for^arbitrary choice of the function f ( y ) . Let b * o. We introduce new variables defined by x = bx - ay y = y then | i = b dx . a c | = b ( a x + f ( y ) ) - a(bx +cy) = -y(ac-bh(y)) "it = at = b x + °y = x ' + a y + °y = x ' + ( a + °) y' 28 T h u s t h e s y s t e m (3.4) r e d u c e s t o t h e f o l l o w i n g d x d t = - y ( a c - b h ( y ) ) (3.7) = x + ( a + c ) y We r e p r e s e n t t h e c u r v e s - y ( a c - b h ( y ) ) = o a n d x + ( a + c ) y = o o b t a i n e d b y p u t t i n g t h e r i g h t h a n d s i d e s o f (3.7) e q u a l t o z e r o o n t h e ( x , y ) p l a n e . T h e f i r s t o f t h e s e r e p r e s e n t s t h e s t r a i g h t l i n e y = o a n d t h e s e c o n d t h e s t r a i g h t l i n e y = a + c . I t i s e a s y t o s e e t h a t dx d t d x d t d t = - y ( a c - b h ( y ) ) > o b e l o w t h e x - a x i s = - y ( a c - b h ( y ) ) < o a b o v e t h e x - a x i s = x + ( a + c ) y > o b e l o w t h e s t r a i g h t l i n e x + ( a + c ) y = o d y . = d t = x + l a + c ) y Co a b o v e t h e s t r a i g h t l i n e x + ( a + c ) y = o T h e d i r e c t i o n o f m o t i o n i s i n d i c a t e d i n f i g . 3. T h e s t r a i g h t l i n e x + ( a + c ) y = o a n d t h e c o - o r d i n a t e a x e s d i v i d e t h e p l a n e ( x , y ) i n t o s i x r e g i o n s . We i n t r o d u c e p o l a r c o - o r d i n a t e s x = r c o s <t> , y = r s i n . T h e n r = x c o s * + y s i n a n d r = - x s i n 4- + f c o s . I t i s n o t d i f  f i c u l t t o v e r i f y t h a t i n d i f f e r e n t r e g i o n s t h e s i g n s o f f a n d tp a r e g i v e n a s b e l o w ; (1,4) r m a y be ^ o, * > o 29 (2,5) r < o , 4> m a y b e \ o (3,6) r m a y b e o , > o . T h e m o t i o n s t a r t e d i n t h e r e g i o n (6) m u s t c r o s s t h e a x i s o f x . T h i s f o l l o w s f r o m t h e i n e q u a l i t y Jf = x + ( a + c ) y > / d > o i n t h e r e g i o n (6). A f t e r e n t e r i n g t h e r e g i o n ( l ) i t c a n n o t r e m a i n t h e r e a n d m u s t c r o s s t h e s t r a i g h t l i n e x + ( a + c ) y = o w i t h t h e i n c r e a s e o f t i m e s i n c e x = I i n t e r s e c t s t h e s t r a i g h t l i n e x + ( a + c ) y = o f o r a l l I. T h e L y a p u n o v f u n c t i o n f o r t h e s y s  t e m (3.7) i s e a s i l y s e e n t o b e 2 yr 2V = x + 2j ( a c - b h ( y ) ) y d y o R e p e a t i n g t h e s a m e a r g u m e n t a s i n T h e o r e m 2.2.1, we a r r i v e a t t h e f o l l o w i n g T h e o r e m 3.1» T h e t r i v i a l s o l u t i o n o f s y s t e m (3.4) i s a s y m p t o t i c a l l y  s t a b l e i n t h e l a r g e u n d e r C o n d i t i o n s (3»5)» I t may b e n o t e d t h a t we m a y o r m a y n o t h a v e * > £ > o i n t h e r e g i o n s ( 1 ) a n d (4). B e f o r e s t u d y i n g t h e c a s e n = 2, k = 1 w e d i s c u s s t h e s y s t e m i - r - » « (3.8) f - - «W T h i s s y s t e m i s a p a r t i c u l a r c a s e o f t h e s y s t e m (2.2.1). S i n c e i t i s q u i t e a n i m p o r t a n t s y s t e m , we d i s c u s s ± t i n d e p e n d e n t l y . T h e c o n d i t i o n s t o w h i c h (3.8) i s s u b j e c t e d a r e x P ( x ) > o , x g ( x ) > o f o r x * o (3.9) P ( o ) = o = g ( o ) 30 T h e V - f u n c t i o n i n t h i s c a s e i s x 2V = y 2 + 2 J g ( x ) d x (3.10) o a n d H = y ( - g ( x ) ) + g ( x ) ( y - P ( x ) ) = - g ( x ) P ( x ) < o f o r x * o = o f o r x = o T h e s t r a i g h t l i n e x = o d o e s n o t c o n t a i n o - p o s i t i v e h a l f t r a j e c t o r y o f t h e s y s t e m (3.8) e x c e p t t h e o r i g i n . I n f a c t J^x = y d t f o r x = o . H e n c e - i t f o l l o w s t h a t t h e t r i v i a l s o l u t i o n i s a s y m p t o t i  c a l l y s t a b l e i n t h e s m a l l . S i n c e t h e r e e x i s t s a L y a p u n o v f u n c t i o n V f o r t h e s y s t e m , t h e r e c a n n o t b e a n y l i m i t c y c l e . A l s o we n o t e t h a t t h e o r i g i n i s t h e o n l y p o i n t o f e q u i l i b r i u m . I t i s n o t d i f f i c u l t t o s e e i n v i r t u e o f t h e C o n d i t i o n s (3-9) t h a t |f = y - P ( x ) > o t o t h e l e f t o f t h e c u r v e y = P ( x ) — = y _ p ( x ) c o t o t h e r i g h t o f t h e c u r v e y = P ( x ) = - g ( x ) ^ o t o t h e l e f t o f t h e y - a x i s = - g ( x ) < o t o t h e r i g h t o f t h e y - a x i s T h e c u r v e y = P ( x ) a n d t h e c o - o r d i n a t e a x e s d i v i d e t h e p l a n e ( x , y ) i n t o s i x r e g i o n s . T h e d i r e c t i o n o f m o t i o n i s r e p r e s e n t e d i n f i g . 4 . We i n t r o d u c e p o l a r c o - o r d i n a t e s x = r c o s <p , y = r s i n 4 . T h e n r = x c o s f> + y s i n • r * = - x s i n + + y c o s * T h e s i g n s o f r a n d i n t h e r e g i o n s a r e g i v e n a s b e l o w : (1,4) r m a y b e ^ o , 4> < o (2 , 5 ) r < o , <fr m a y b e >^L o (3 ,6) r m a y b e ^ o , 4> < o . 3 1 L e t u s f o l l o w t h e m o t i o n M ( t ) a f t e r i t i n t e r s e c t s t h e n e g a  t i v e h a l f y - a x i s . L e t u s a s s u m e t h a t y = A i n t e r s e c t s t h e c u r v e y = F ( x ) f o r a l l A , i . e . , F ( x ) = A h a s a s o l u t i o n f o r a l l A, t h e n a s i n c e i n t h e r e g i o n ( 4 ) 4» < o a n d y i s i n c r e a s i n g t h e m o t i o n M ( t ) c a n n o t r e m a i n i n t h e b o u n d e d r e g i o n o L M . I t d e f i n i t e l y c a n n o t c r o s s F I G . 4 L M a n d o L . T h e r e f o r e i t m u s t c r o s s t h e c u r v e y = F ( x ) a n d e n t e r t h e r e g i o n ( 5 ) . I n t h e r e g i o n (5) i t e i t h e r e n t e r s t h e o r i g i n o r t h e r e g i o n ( 6 ) . T h i s f o l l o w s f r o m t h e f a c t t h a t r < o i n t h i s r e g i o n . S i n c e = - F ( x ) + y >/ d > o i n t h e r e g i o n ( 6 ) , t h e d t m o t i o n m u s t l e a v e t h i s r e g i o n a n d e n t e r t h e r e g i o n ( l ) . S i m i l a r a r g u m e n t h o l d s f o r t h e r e g i o n s ( l ) , ( 2 ) a n d ( 3 ) . T h i s s h o w s t h a t t h e r e a r e n o m o t i o n s w i t h b o u n d e d p o l a r a n g l e s i n t h e r e g i o n s ( l ) , ( 3 ) , ( 4 ) a n d ( 6 ) . T h e m o t i o n s w i t h b o u n d e d p o l a r a n g l e s c a n o n l y o c c u r i n t h e r e g i o n s ( 2 ) a n d ( 5 ) , a n d a s p r o v e d a b o v e t h e y a r e b o u n d e d . 'Thus i t f o l l o w s t h a t a l l m o t i o n s w i t h b o u n d e d p o l a r a n g l e s a r e b o u n d e d . F o x ' t h e s t r a i g h t l i n e L ( o , 00) c a n b e t a k e n t h e p o s i  t i v e h a l f y - a x i s . T h u s a l l t h e c o n d i t i o n s o f T h e o r e m 1 . 2 . 7 a r e s a t i s f i e d a n d w e h a v e t h e f o l l o w i n g t h e o r e m ! T h e o r e m 3 . 2 . I f t h e C o n d i t i o n s ( 3 » 9 ) a r e s a t i s f i e d a n d F ( x ) = A  h a s a s o l u t i o n f o r a l l A, t h e n t h e t r i v i a l s o l u t i o n o f (3.8) i s 32 a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . We r e m a r k t h a t b y u s i n g T h e o r e m 1.2.5 t h e s t a b i l i t y i n t h e l a r g e h o l d s i f x J"g(x) d x —* » a s | x | —* oo o . T h i s i s r e q u i r e d i n o r d e r t o m a k e t h e V f u n c t i o n i n (3.10) i n  f i n i t e l y l a r g e . T h e w i d e l y d i s c u s s e d e q u a t i o n 2 + f ( x ) d x + g ( x ) = o (3.11) at* dt c a n b e d e a l t i n t h e same w a y a s t h e s y s t e m (3.8), s i n c e (3.1l) c a n b e t r a n s f o r m e d t o X Jf = g ( x ) w h e r e P ( x ) = j" f ( x ) d x o WTe s h a l l n o w d i s c u s s t h e Aizerman p r o b l e m f o r t h e c a s e n = 2, k = 1 , i . e . , t h e s y s t e m If = f U) + a y (3.12) f t = b x + C y u n d e r t h e c o n d i t i o n s : f ( o ) = o ; c + h ( x ) < o j c h ( x ) - a b > o (3.13) for x + o w h e r e h ( x ) = f ( x ) . x * o x T h i s s y s t e m i s d i s c u s s e d i n t h e w o r k s o f E r u g i n ( t 8 j , [13] ) a n d M a l k i n [2]^ . I t w a s p r o v e d b y E r u g i n t h a t i f c ~ + a b t- o , t h e n 33 t h e t r i v i a l s o l u t i o n o f (3.12) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r c o n d i t i o n s (3.13). T h e p r o o f b y E r u g i n o f t h i s t h e o r e m i s q u i t e l e n g t h y , s o we g i v e h e r e a s h o r t p r o o f o f t h e t h e o r e m . I t w i l l b e c o m e c l e a r f r o m o u r p r o o f w h y t h e s t a b i l i t y i n t h e l a r g e d o e s n o t h o l d u n d e r c o n d i t i o n s (3.13) i n t h e c a s e c + a b = o . E r u g i n a l s o s h o w e d t h a t f o r t h e s t a b i l i t y i n t h e l a r g e i n t h e c a s e 2 c + a b = o , i t i s s u f f i c i e n t t h a t l i m x - » < * J ( c f ( x ) - a b x ) d x + c f ( x ) - a b x 1= + r (3.14) l i m x -j x —> - <*> j ( c f ( x ) - a b x ) d x - c f ( x ) + a b x L + «° . o J K r a s o v s k i i (17) s h o w e d t h a t c o n d i t i o n s (3.14) a r e n e c e s s a r y a s w e l l a s s u f f i c i e n t . T h e q u e s t i o n o f t h e r e g i o n o f s t a b i l i t y i n t h e c a s e w h e r e t h e s t a b i l i t y i n t h e l a r g e d o e s n o t h o l d i s d i s c u s s e d b y P l i s s [25]. 2 We a s s u m e t h a t c + a b * o . L e t a = o , t h e n i m m e d i a t e i n t e g r a t i o n o f t h e s y s t e m y i e l d s t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n x = o = y o f s y s t e m (3.12). L e t n o w a * o . We i n  t r o d u c e n e w d e p e n d e n t v a r i a b l e s x = x y = a y - c x + y + c x T h e n If- = dx = f(x') ' ' d t dt | £ = a % - c d x = a ( b x + c x ) - c ( f ( x ) + a y ) a t d t d t - a b x - c f ( x ) 34 The system (3.12) i s reduced to the form H = f(x) + cx + y § = - x(c h(x) - ab) Comparing, i t with system (3-8) w ® have F(x) = - x(c + h(x)) and g(x) = x( c h(x) - ab) The curve y + cx + f(x) = o and the co-ordinate axes divide the (x,y) plane into six regions. The direction of motion i s repre sented in fig . 5- We consider the following cases. Case 1 c < o. Consider the straight line 2 , y = - ° + a " x (3.15) If c + ab > o, then the straight line (3-13) l i e s i n the f i r s t and third quadrants. Let us see how i t i s situated on the (x,y) plane with respect to the curve y + cx + f(x) = o (3.16) r I G . 5 Let y( 1 ) and y^ 2) denote the ordinates of (3.16) and (3.15) respectively, then (1 ) (2) 2 y - y = - cx - f(x) + £ t ab x c 2 2 = -c x - cf(x) +{c + ab) x = -x(c h(x) - ab) ^  0 c c according as x ^ o , 35 i . e . , t h e c u r v e (3.16) l i e s a b o v e t h e s t r a i g h t l i n e (3.15) i n t h e f i r s t q u a d r a n t a n d b e l o w i n t h e t h i r d q u a d r a n t . . O b v i o u s l y , t h e 2 s t r a i g h t l i n e s y = A a n d y = - c + a P x i n t e r s e c t , w h e n c e f o l l o w s c t h a t y = A a n d t h e c u r v e y = - c x - f ( x ) i n t e r s e c t f o r a l l A . H e n c e t h e t r i v i a l s o l u t i o n o f (3.12) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e i n t h i s c a s e a c c o r d i n g t o T h e o r e m 3.2. 2 2 I f c + ab < o , t h e n t h e s t r a i g h t l i n e y = - c + a p x c l i e s i n t h e s e c o n d a n d f o u r t h q u a d r a n t s a n d n o s u c h c o n c l u s i o n a s a b o v e c a n b e d r a w n . H o w e v e r , w e c a n p r o v e t h a t i n t h i s c a s e <j> < - 6 < o i n t h e r e g i o n s ( l ) a n d (4) w h i c h e n s u r e s t h e s t a b i l  i t y i n t h e l a r g e . A c c o r d i n g t o C o n d i t i o n s (3.13) c + h ( x ) < o , c h ( x ) - a b > o f o r x * o i . e . h ( x ) < - c , h ( x ) < c 2 b u t s i n c e c + a b < o , t h e a b o v e i n e q u a l i t i e s a r e s a t i s f i e d i f we t a k e h ( x ) < - c - f ( x ) c a n t h e n b e w r i t t e n a s f ( x ) = - c x - J . (x) w h e r e t ( x ) Jfc. o a c c o r d i n g a s X o L e t u s c a l c u l a t e <<> : r 4> = s i n < * » ( f ( x ) + c x + y ) - c o s ^ > ( c h ( x ) - a b ) x = - s i n 4 > ( y - X » ( x ) ) - c o s * £ c ( - c x - £(x)) - a b x ] = - s i n < ^ ( y - X » ( x ) ) + c o s 4> £ ( c ^ + a b ) x + c L ( x ) j * = - S i n * ( y - X.(X)) + C O S * [ ( c2 + a b ) x + ^ . € < o i n t h e r e g i o n s ( l ) a n d (4). 36 2 Case 11 c > o. In this case c + ab < o,because from Conditions (3.13) < h(x) C - c c Let h(x) be taken as h(x) = - c - JC(x) ^ ^ < - c - i ( x ) < - c c c + a b < _ x ( x ) < o c - c 2+ ab > x. ( x) > o c or o < JC (x) <. - ° 2 + a b c Let us consider the region ( l ) . If <t> £ - £ < o, then i t means that - l i m -~[(c2+ ab) x + cxX(x)l = o, i.e. l i m xjC(x) = <*, X—*<*> L J x- whence i t follows that the straight lin e y = A intersects the curve y = - cx - f(x) for a l l A In fact, - A = cx + f(x) = cx + (-cx - xX(x)) = - x-X(x) therefore x £(x) = A If x £(x) = A does not have a solution for a l l A, then 1 ™ x<(x) = f i n i t e = D x— lim J(c2+ a b ) x + cxX(x ) l^li!H f-(c2+ ab) x] + l i E (_c xjl(x)) X — = l i E [-x(c 2 + ab)] - ( C X JC(x)) , X - » « o ~ X whence follows that <p <, - fi < o. Case 111 c = o. The System (3.12) reduces to 37 £ • ••»« a n d t h e C o n d i t i o n s (3.13) r e d u c e : t o h ( x ) < o , a b < o . P o r t h i s s y s t e m w e t a k e t h e f o l l o w i n g L y a p u n o v ' • f u n c - 2 2 t i o n , 2V = - a b x + y t h e d e r i v a t i v e o f w h i c h i s , 2. / \ — = - a b x h ( x ) T h i s V f u n c t i o n s a t i s f i e s a l l t h e c o n d i t i o n s o f T h e o r e m 1.2.5. C o m  b i n i n g a l l t h e s e r e s u l t s we h a v e t h e f o l l o w i n g t h e o r e m : T h e o r e m 3.3. I f f o r t h e S y s t e m (3.12) c2+ a b * o , t h e n t h e t r i v i a l  s o l u t i o n i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r C o n d i t i o n s  (3.13). 2 I n t h e c a s e c + a b = o , we r e m a r k t h a t c < o a n d f r o m t h e p r o o f o f s t a b i l i t y i n t h e C a s e 1, w e f i n d t h a t t h e s t r a i g h t l i n e y = - ° 2 + a D x c o i n c i d e s w i t h t h e x - a x i s a n d h e n c e we c a n c n e i t h e r s a y t h a t t h e s t r a i g h t l i n e y = A i n t e r s e c t s t h e c u r v e y + c x + f ( x ) = o f o r a l l A n o r 4, ^ . ^ ^ o i n t h e r e g i o n s ( l ) a n d (4). 38 4. A g e n e r a l i z a t i o n o f t h e p r o b l e m o f A i z e r m a n . I n t h i s s e c t i o n w e d i s c u s s t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s g i v e n b y d x d t a x + ^ ( y ) (4.1) d t f 2 ( x ) + c y u n d e r t h e c o n d i t i o n s : w h e r e h . ( y ) a n d h „ ( x ) a r e d e f i n e d b y f . ( y ) = y h , ( y ) a n d f 9 ( x ) = x h 9 ( x ) . o b t a i n e d c e r t a i n t h e o r e m s r e g a r d i n g t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n x = o = y o f (4.1) . We p r o v e h e r e t h e f o l l o w i n g t h e o r e m : T h e o r e m 4.1. I f e i t h e r h ^ ( y ) > o f o r y $ o a n d h 2 ( x ) > o f o r x $ o o r h ^ ( y ) < o f o r y * o , h ^ ( x ) < o f o r x * o , t h e n t h e t r i v i a l s o l u - t i o n o f ( 4 » l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r C o n d i t i o n s P r o o f . We a s s u m e h ^ ( y ) > o , y * o , a n d h g ( x ) > o , x * o . P r o m C o n  d i t i o n s (4.2) f o l l o w s t h a t a < o , c < o . We c o n s i d e r t h e c u r v e s t h e f i r s t a n d t h i r d q u a d r a n t s . T h e same i s t r u e f o r t h e c u r v e f ^ ( x ) + c y = o . We f u r t h e r n o t e t h a t t h e c u r v e a x + f ^ ( y ) = o l i e s a b o v e t h e c u r v e f p ( x ) + c y = o i n t h e f i r s t q u a d r a n t a n d b e l o w i n T h e a b o v e s y s t e m w a s f i r s t d i s c u s s e d b y K r a s o v s k i i [ l & j w h o a x + f.j ( y ) = o a n d f 2 ( x ) + c y = o (4.3) S i n c e h . ( y ) > o , y + o a n d a < o , t h e c u r v e a x + f . ( y ) = o l i e s i n 39 t h e t h i r d q u a d r a n t . I n f a c t , i f y ^ a n d y 2 d e n o t e t h e o r d i n a t e s o f t h e c u r v e s i n (4.3) r e s p e c t i v e l y , t h e n *1 " y2 - - a x ^ ( y 1 ) h 2 ( x ) a c - to i ( y ) h ? ( x ) - x !— f N — ^ o a c c o r d i n g a s c \ ( y ) x >• o I t i s e a s y t o s e e t h a t § = a x + f ^ y ) > o f t = a x + f 1 ( y ) < o § = f 2 ( x ) + c y < o | £ = f 2 ( x ) + c y > o t o t h e l e f t o f t h e c u r v e a x + f ^ ( y ) = o t o t h e r i g h t o f t h e c u r v e a x + f ^ ( y ) = o t o t h e l e f t o f t h e c u r v e c y + f r , ( x ) = o t o t h e r i g h t o f t h e c u r v e c y + f 2 ( x ) = o T h e f u n c t i o n x ( t ) i s m a x i m u m o n t h e c u r v e a x + f ^ ( y ) = o f o r y > o a n d m i n i m u m f o r y < o ; a n d t h e f u n c t i o n s y ( t ) i s m a x i m u m o n t h e c u r v e f 2 ( x ) + c y = o f o r y > o a n d m i n i m u m f o r y < o . T h e c u r v e s a n d t h e d i r e c t i o n o f m o t i o n a r e s h o w n i n f i g . 6 . A s b e f o r e we i n t r o d u c e p o l a r c o - o r d i n a t e s x = r c o s <p , y = r s i n $ . T h e n r = x cos«fr + y s i n * r * = - x s i n & + y c o s <p T h e s i g n s o f f a n d 4> i n d i f f e r e n t r e - / g i o n s a r e ( l,5) r m a y b e ^ . o , <. o (2,6) r < o , m a y b e >^o (3.7) r m a y b e >y<i o , <* > o (4.8) r < o , 4> m a y b e o P r o m T h e o r e m 2.1 o f E r u g i n ' s w o r k [8] i t f o l l o w s t h a t t h e r e 40 i s a t l e a s t o n e i n t e g r a l c u r v e g o i n g t o t h e o r i g i n , t h e o n l y p o i n t o f e q u i l i b r i u m , i n e a c h o f t h e r e g i o n s (4) a n d (8). O t h e r m o t i o n s s t a r t e d i n t h e r e g i o n s (4) o r (8) e i t h e r g o t o t h e o r i g i n o r e n t e r t h e r e g i o n s (3), (5) o r ( l ) , (8) ( s i n c e r < o , 4> m a y b e >^o i n (4) a n d (8)). L e t u s s u p p o s e t h a t t h e m o t i o n e n t e r s t h e r e g i o n ( l ) . We s h o w t h a t t h e m o t i o n l e a v e s t h i s r e g i o n w i t h t h e i n c r e a s e o f t i m e . L e t c 2 b e t h e l e a s t u p p e r b o u n d o f h 2 ( x ) . c 2 i s f i n i t e o t h e r w i s e a c - ( y ) h 2 ( x ) > o w i l l n o t h o l d . We c o n s i d e r t h e s t r a i g h t l i n e CgX + c y = o . I t i s e a s y t o v e r i f y t h a t t h i s s t r a i g h t l i n e l i e s a b o v e t h e c u r v e f 2 ( x ) + c y = o a n d b e l o w t h e c u r v e a x + f ^ ( y ) = o „ i n t h e f i r s t q u a d r a n t . I n t h e t h i r d q u a d r a n t t h e p o s i t i o n s a r e r e  v e r s e d . We c o n s i d e r t h e r e g i o n b o u n d e d b y y = L, t h e c u r v e a x + f ^ ( y ) = o a n d t h e p o s i t i v e h a l f y - a x i s . T h e m o t i o n c a n n o t i n  t e r s e c t t h e s t r a i g h t l i n e y = t,since $2. < o . I t c a n n o t g o t o t h e d t o r i g i n , s i n c e <t> <. o . T h e r e f o r e i t m u s t i n t e r s e c t t h e c u r v e a x + f ^ ( y ) = o a n d e n t e r t h e r e g i o n (2) w h e r e i t g o e s t o t h e o r i  g i n w i t h t h e i n c r e a s e o f t i m e . P o r r e g i o n (3) we t a k e t h e s t r a i g h t l i n e x = t a n d s i m i l a r l y s h o w t h a t t h e m o t i o n c r o s s e s t h e c u r v e f 2 ( x ) + c y = o a n d e n t e r s t h e r e g i o n (2) w h e r e i t g o e s t o t h e o r i g i n a s t—» + oo . S i m i l a r r e a s o n i n g s h o l d f o r t h e r e g i o n s (5) a n d (7) w h i c h c o m p l e t e s t h e p r o o f o f T h e o r e m 4.1. I n t h e same w a y w e c a n p r o v e t h e f o l l o w i n g - t h e o r e m f o r t h e s y s t e m (4.4) TjJ = b x + c y 41 voider the conditions: h.j(x) + c<o, x * o, ch^x) - h 2(y) b > o, x * o, y * o, (o)=f 2(o)=o > where h (x) = 1^1 , x * o, h 2(y) = Mil , y * o (4.5) x y Theorem 4.2. If b h g(y) > o for y * o and the Conditions (4.5) are satisfied, then the t r i v i a l solution of system (4.4) i s asymptoti  cally stable in the large. 42 5. T h e s t a b i l i t y i n t h e l a r g e o f t h e s y s t e m Jf = F ( x , y ) , Jf = f (•") u s i n g q u a l i t a t i v e m e t h o d s . I n t h i s s e c t i o n we c o n s i d e r t h e s y s t e m ( 2 . l ) . T h e a s s u m p  t i o n s u n d e r w h i c h we w i l l b e w o r k i n g w i l l b e d i f f e r e n t f r o m t h o s e o f S e c t i o n 2. B . A . E r s h o v ( 6 ] d i s c u s s e d t h i s s y s t e m a n d o b t a i n e d c e r  t a i n t h e o r e m s r e g a r d i n g t h e s t a b i l i t y i n t h e l a r g e o f t h e t r i v i a l s o l u t i o n . We h e r e s h o w t h a t h i s r e s u l t s a r e c o r r e c t b u t t h e p r o o f s a r e w r o n g . L e t u s c o n s i d e r t h e s y s t e m (5.1) Jf = f(«0 , w h e r e «- = c.,x - d . , y a n d c^ , d^ a r e p o s i t i v e c o n s t a n t s . We s h a l l a s s u m e t h a t F ( x , y ) i s a c o n t i n u o u s f u n c t i o n , h a v i n g f i r s t o r d e r p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o x a n d y , f o r a l l v a l u e s o f x , y . F u r t h e r we a s s u m e t h a t 1| * o , F ( o , o ) = o (5.2) We s h a l l d i s c u s s t h r e e c a s e s ( i ) | ^ < ° j ( i i ) 5 Z = o , v / 3x -&X ( i i i ) 4 1 > o . ^x T h e c o n t i n u o u s f u n c t i o n f ( « " ) , a p p e a r i n g i n t h e r i g h t h a n d s i d e o f t h e s e c o n d e q u a t i o n o f ( 5 « l ) i s s u b j e c t e d t o t h e f o l l o w i n g c o n d i t i o n s c f ( « * ) > o f o r «- $ o (5.3) f ( o ) = o (5.4) >/ o f o r a l l (5.5) 9 6 * 43 I n t h e c o n d i t i o n (5.5) 2i£ i s n o t t a k e n t o b e i d e n t i c a l l y as- z e r o . The e q u a t i o n s (5.1) c a n b e w r i t t e n a s § -(-if) x + (-rf) y + ^ ' x = o , y = o \°J J x = 0 ) y = 0 \ ° / x = o , y = o \ »«V x = o , y = o d t Q w h e r e Mx.y) = F(z,y) -(^f ) x - ( ^ | ) y ^ X ' x = o , y = o V ' y ' x = c , y = „ ( x, y ) . ,(0-.^) « . a, ( O f ) y x = o = y We w r i t e f o r s i m p l i c i t y ' x = o = y = a , \ * y ' x = o = y = b '1 x = o , y = o x 0 / ; = d x = o = y T h e s y s t e m (5*6) c a n t h e n b e w r i t t e n i n t h e f o r m d x d t = - N a x - b y + °y ( x , y ) |f = c x - dy + <y ( x , y ) (5.6) (5.7) T h e t h r e e c a s e s t h u s c o r r e s p o n d t o t h e v a l u e s 1, o , - 1 o f N i n (5.7) r e s p e c t i v e l y . C a s e 1 _2LZ < o . P r o m e q u a t i o n s (5.7) we h a v e f o r N = 1 a x |f = - a x - b y + % ( x , y ) |f = c x - d y + «H ( x , y ) (5.8) 44 T h e e q u a t i o n s o f t h e f i r s t a p p r o x i m a t i o n a r e d x , — = - a x - b y (5.9) Jf = c x - d y T h e c h a r a c t e r i s t i c e q u a t i o n o f (5«9) i s A + a - b = o c Ti+ d 2 o r ^ + ( a + d ) h + a d + b e = o (5.10) ' T h e r o o t s o f t h i s e q u a t i o n h a v e n e g a t i v e r e a l p a r t s , s i n c e a + d > o a n d a d + b e > o S i n c e t h e r o o t s o f (5.10) h a v e n e g a t i v e r e a l p a r t s , i t f o l l o w s t h a t t h e t r i v i a l s o l u t i o n x = o = y o f ( 5 . 8 ) i s a s y m p t o t i c a l l y s t a b l e a c  c o r d i n g t o L y a p u n o v . T h e a b s e n c e o f p e r i o d i c s o l u t i o n s i s e a s i l y s e e n b y u s i n g t h e c r i t e r i o n o f B e n d i x s o n . I n f a c t , M. + 1L _lZ _ d JLL 4 0 j x ny i x 1 i n v i e w o f c o n d i t i o n (5*5) a n d t h e f a c t t h a t JLE. < O , d , > o . 3 x 1 We r e p r e s e n t t h e c u r v e s P ( x , y ) = o a n d f ( • * ) = o o n t h e ( x , y ) p l a n e . B y v i r t u e o f C o n d i t i o n s (5.5) a n d (5.4) f ( ** ) = o r e p r e  s e n t s t h e s t r a i g h t l i n e «* = o . S i n c e c^ a n d d ^ a r e p o s i t i v e c o n  s t a n t s , f = o i s s i t u a t e d i n t h e f i r s t a n d t h i r d q u a d r a n t s . T h e c u r v e P ( x , y ) = o p a s s e s t h r o u g h t h e o r i g i n s i n c e P ( o , o ) = o . I t i s s i t u a t e d i n t h e s e c o n d a n d f o u r t h q u a d r a n t s s i n c e t h e s l o p e o f P ( x , y ±'e'> & = *^\x i s n e g a t i v e , d x -Tjyfc T h e s e c o n s i d e r a t i o n s s h o w t h a t ( o , o ) i s t h e o n l y p o i n t o f 45 e q u i l i b r i u m o f (5.8). We h a v e s h o w n s o f a r t h a t ( i ) t h e o r i g i n i s t h e o n l y p o i n t o f e q u i l i b r i u m , ( i i ) i t i s a s y m p t o t i c a l l y s t a b l e i n t h e s e n s e o f L y a p u n o v , ( i i i ) t h e r e e x i s t . n o p e r i o d i c s o l u t i o n s . We s h a l l n o w s h o w t h a t t h e r e e x i s t s a s t r a i g h t l i n e L ( o , <* ) w h i c h i s i n t e r s e c  t e d b y t h e m o t i o n s i n o n e d i r e c t i o n o n l y a n d a l l m o t i o n s w i t h b o u n d e d p o l a r a n g l e s a r e b o u n d e d . W i t h t h i s t h i n g i n v i e w we e x a m i n e t h e d i r e c  t i o n s o f m o t i o n s ( s e e f i g . 7 ) . I t i s e a s y t o s e e t h a t |f = P ( x , y ) > o f o r t h e p o i n t s , l y i n g t o t h e l e f t o f t h e c u r v e P ( x , y ) = o d x — = P ( x , y ) <o f o r t h e p o i n t s , l y i n g t o t h e r i g h t o f t h e c u r v e P ( x , y ) = o |f = f(o* ) >o f o r t h e p o i n t s , l y i n g b e l o w t h e s t r a i g h t l i n e = o |f = f ( c ) < o f o r t h e p o i n t s , l y i n g a b o v e t h e s t r a i g h t l i n e f = o T h e f u n c t i o n y ( t ) a t t a i n s m a x i m u m f o r y > o a n d m i n i m u m f o r y < o o n t h e s t r a i g h t l i n e f ( « " ) = o ; x ( t ) a t t a i n s m a x i m u m f o r x > o a n d m i n i m u m f o r x < o o n t h e c u r v e F ( x , y ) = o . T h e s t r a i g h t l i n e f(c) = o , t h e c u r v e P ( x , y ) = o a n d t h e c o - o r d i n a t e a x e s d i v i d e t h e ( x , y ) p l a n e i n t o e i g h t r e g i o n s . We i n t r o d u c e FIG.7 46 p o l a r c o - o r d i n a t e s x = r c o s <$ , y = r s i n <p .. T h e n r = x c o s + y s i n <P a n d r = - x s i n <p + y c o s <j> t h e s i g n s o f r a n d <p i n d i f f e r e n t r e g i o n s a r e a s f o l l o w s : ( l,5) r m a y h e ^ o , > o (2,6) r < o , <j> m a y b e ^ o (3,7) r m a y b e ^ o , <p > o (4,8) r < o , m a y b e >^ o . E r s h o v [6} a r g u e d t h a t , s i n c e qi > o i n t h e r e g i o n s ( l ) , (3), (5) a n d (7) t h e r e c a n n o t b e a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i n t h e s e r e g i o n s a n d a n y m o t i o n f a l l i n g i n t h e s e r e g i o n s o r s t a r t i n g t h e r e h a s t o g e t o u t o f t h e s e r e g i o n s a f t e r i n t e r s e c t  i n g e i t h e r t h e s t r a i g h t l i n e f ( « r ) = o o r t h e c u r v e F ( x , y ) = o . T o u s t h i s r e a s o n i n g i s d o u b t f u l , s i n c e 4> > o i s n o t s u f f i c i e n t t o g u a r a n t e e t h a t t h e r e c a n n o t b e a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i n t h e s e r e g i o n s . H o w e v e r , t h e a b o v e a s s e r t i o n r e m a i n s t r u e i f we c o u l d s h o w t h a t <p > 5 > o i n t h e s e r e g i o n s . I n f a c t , i f «i > <t , t h e n i n t e g r a t i n g we h a v e <P - <*>q > 6 ( t - t ) , w h e n c e f o l l o w s t h a t a s t i n c r e a s e s , i n c r e a s e s a n d h e n c e t h e r e w i l l b e a n i n  s t a n t o f t i m e w h e n t h e m o t i o n l e a v e s t h e s e r e g i o n s . L e t a m o t i o n M ( t ) a f t e r i n t e r s e c t i n g t h e n e g a t i v e h a l f y - a x i s e n t e r t h e r e g i o n (7). We s h o w t h a t m o t i o n M ( t ) c a n n o t r e  m a i n i n t h i s r e g i o n a n d c o n s e q u e n t l y i t w i l l e n t e r t h e r e g i o n (8). T o s h o w t h i s we w r i t e <$> = J_ f - x s i n « f > + y c o s * ] r 1 1 (5.11) = — ( - P ( x , y ) s i n <> + f(cy) c o s « > ) > o i n r e g i o n (7) 47 If M ( t ) d o e s n o t c r o s s t h e c u r v e F ( x , y ) = o , t h e n x ( t ) " b e c o m e s i n f i n i t e s i n c e x ( t ) a n d y ( t ) a r e i n c r e a s i n g a n d £ > o . F o r s u f  f i c i e n t l y l a r g e x , w i t h t h e i n c r e a s e o f x , <r i n c r e a s e s . S i n c e i t i s a s s u m e d t h a t _]!_£. >, o , f(v) i s n o n d e c r e a s i n g a n d t h e r e f o r e f r o m ( 5 . 1 l ) i t f o l l o w s t h a t «j> c a n a l w a y s b e t a k e n g r e a t e r t h a n $ > o a n d h e n c e w i t h t h e i n c r e a s e o f t i m e t h e m o t i o n m u s t l e a v e t h e r e g i o n ( 7 ) , w h i c h c o n t r a d i c t s o u r a s s u m p t i o n . A f t e r e n t e r i n g t h e r e g i o n (8) i t e i t h e r g o e s t o t h e o r i g i n a s t — » + °© o r i n  t e r s e c t s t h e x - a x i s a n d e n t e r t h e r e g i o n ( l ) . T h i s f o l l o w s f r o m t h e f a c t t h a t i n t h i s r e g i o n f < o a n d tp m a y b e \ o . T h e m o t i o n c a n n o t r e m a i n i n t h e r e g i o n ( l ) a n d m u s t e n t e r t h e r e g i o n (2). T o s e e t h i s , c o n s i d e r t h e r e g i o n b o u n d e d b y t h e s t r a i g h t l i n e <r* = o, x = A a n d t h e x - a x i s . S i n c e i n t h e r e g i o n ( l ) x d e c r e a s e s , t h e m o t i o n c a n n o t i n t e r s e c t t h e l i n e x = A . I t c a n n o t g o t o t h e o r i g i n ^ s i n c e 4> > o a n d t h e r e f o r e i t m u s t n e c e s s a r i l y g o o u t o f t h e r e  g i o n ( l ) a n d e n t e r t h e r e g i o n (2). H e r e > i . e . J i n r e g i o n ( 2 ) , s i n c e r <£ o t h e m o t i o n e i t h e r g o e s t o t h e p o i n t o f e q u i l i b r i u m o r e n t e r s t h e r e g i o n (3). S i m i l a r r e a s o n i n g h o l d s f o r t h e r e s t o f t h e r e g i o n s . A l l t h i s s h o w s t h a t a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d . P o r t h e s t r a i g h t l i n e L ( o , ° o ) we c a n t a k e t h e p o s i t i v e h a l f y - a x i s . T h u s a l l t h e c o n d i t i o n s o f T h e o r e m 1.2 . 7 a r e s a t i s f i e d a n d we h a v e t h e f o l l o w i n g t h e o r e m : T h e o r e m 5 . 1 . L e t J|Z <_ o . T h e n u n d e r c o n d i t i o n s (5.2) - (5.5) 0 x  t h e t r i v i a l s o l u t i o n o f ( 5 . l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . 48 C a s e I I U! = o . T h e e q u a t i o n s (5.7) i n t h i s c a s e t a k e t h e f o r m ti x & - - * • -sty) Jf = c x - d y + ~i> ( x , y ) T h e e q u a t i o n s o f f i r s t a p p r o x i m a t i o n a r e d x , d t = " b y Jf = c x - d y (5.12) (5.13) T h e c h a r a c t e r i s t i c e q u a t i o n o f (5.13) i s g i v e n b y h - b = o o r f, + d ? i + b c = o c \ + d T h e r e a l p a r t s o f t h e r o o t s o f t h i s e q u a t i o n a r e n e g a t i v e , s i n c e d > o a n d b e > o , w h e n c e f o l l o w s t h e a s y m p t o t i c s t a b i l i t y o f t h e t r i v i a l s o l u t i o n o f s y s t e m (5.12) i n t h e s e n s e o f L y a p u n o v . T h e c u r v e F ( y ) = o r e p r e s e n t s t h e s t r a i g h t l i n e y = o a n d f ( «•*) = o r e p r e s e n t s t h e s t r a i g h t l i n e w = o . T h e o r i g i n i s t h e o n l y common p o i n t o f f ( c ) = o . a n d F ( y ) = o . I t i s e a s y t o s e e t h a t cLx -rr = ^ ( y ) > o f o r t h e p o i n t s , l y i n g b e l o w t h e x - a x i s Q.T/ — = P ( y ) < o f o r t h e p o i n t s , l y i n g a b o v e t h e x - a x i s CLX = f ( c ) > o f o r t h e p o i n t s , b e l o w t h e s t r a i g h t l i n e f ( ° * ) = o d t i •^7 = f ( c ) < o f o r t h e p o i n t s , a b o v e t h e s t r a i g h t l i n e f (**)= o CI t T h e f u n c t i o n y ( t ) i s m a x i m u m f o r y > o a n d m i n i m u m f o r y < o f o r t h e p o i n t s o n f (<**) = o a n d x ( t ) i s m a x i m u m f o r x > -o- a n d 49 m i n i m u m f o r x < o f o r t h e p o i n t s o n t h e x - a x i s . T h e c u r v e s F ( y ) = o , f («•*) = o a n d t h e a x e s o f c o - o r d i n a t e s d i v i d e t h e p l a n e i n t o s i x r e g i o n s . T h e d i r e c t i o n o f m o t i o n i s r e p r e  s e n t e d i n f i g . 8. T h e s i g n s o f r a n d 4> i n d i f f e r e n t r e g i o n s a r e g i v e n a s b e  l o w : (1.4) r m a y b e ^ 0 , <*> > o (2.5) r < o , <p m a y b e >^ 0 (3.6) r m a y b e " ^ o , d> > o I n t h e r e g i o n (6) i t i s e a s y t o s e e t h a t >, £ > o , w h e n c e i t f o l l o w s t h a t t h e r e e x i s t s a n i n - d t s t a n t t w h e n t h e m o t i o n i n t e r s e c t s t h e x - a x i s . A s i n C a s e I i t c a n b e s h o w n t h a t t h e m o t i o n M ( t ) i n t e r s e c t s t h e s t r a i g h t l i n e f(«0 = o a n d e n t e r s t h e r e g i o n (2) w i t h t h e i n c r e a s e o f t i m e . I n r e g i o n (2) e i t h e r t h e m o t i o n g o e s t o t h e o r i g i n w i t h t h e i n c r e a s e o f t i m e o r e n t e r s t h e r e g i o n (3). S i m i l a r a r g u m e n t s c a n b e a p p l i e d f o r t h e r e g i o n s (3),(4) a n d ( 5 ) . T h u s we h a v e s h o w n t h a t ( i ) (0,0) i s t h e o n l y p o i n t o f e q u i  l i b r i u m , ( i i ) i t i s a s y m p t o t i c a l l y s t a b l e i n t h e s m a l l , ( i i i ) a n y m o  t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d . T o s h o w t h a t t h e r e e x i s t s n o p e r i o d i c m o t i o n s , we u s e t h e c r i t e r i o n o f B e n d J x s o n F > f -j -i^> / N — — + — — = o - d j » i_ ^ o ( n o t i d e n t i c a l l y e q u a l . t o z e r o ) d X } y - j ^ . F o r t h e s t r a i g h t l i n e L(o,oo) we c a n t a k e t h e p o s i t i v e h a l f x - a x i s . T h u s a l l t h e c o n d i t i o n s o f T h e o r e m 1.2.7 a r e s a t i s f i e d a n d we h a v e t h e f o l l o w i n g t h e o r e m ; 50 T h e o r e m 5.2. L e t ^ = o . T h e n u n d e r c o n d i t i o n s (5.2) - (5.5) t h e o x t r i v i a l s o l u t i o n o f s y s t e m ( 5 . l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e  l a r g e . C a s e I I I !LE > o . T h e e q u a t i o n s (5.7) i n t h i s c a s e t a k e t h e f o r m d x = a x - b y + - \ ( x , y ) d t (5.14) J&L = c x - d y + <y ( x , y ) d t T h e c h a r a c t e r i s t i c e q u a t i o n o f t h e f i r s t a p p r o x i m a t i o n i s ^ - a - b = o c a + d o r A + ( d - a ) * + b e - a d = o (5.15) T h e r o o t s o f (5.15) w i l l h a v e n e g a t i v e r e a l p a r t s u n d e r t h e f o l l o w  i n g c o n d i t i o n s ; d - a > o (5.16) b e - d a > o (5.17) I f t h e c o n d i t i o n s (5.16) a n d (5.17) a r e s a t i s f i e d t h e n t h e t r i v i a l s o l u t i o n x = o = y o f (5.14) i s a s y m p t o t i c a l l y s t a b l e i n t h e s m a l l . We n o w r e p r e s e n t t h e c u r v e s F ( x , y ) = o a n d t h e s t r a i g h t l i n e f ( o ' ) = o o n t h e ( x , y ) p l a n e . S i n c e _LZ > o , t h e c u r v e F ( x , y ) = o i s 3 x s i t u a t e d i n t h e f i r s t a n d t h i r d q u a d r a n t s a n d s o i s t h e s t r a i g h t l i n e g i v e n b y f ( « * ) = o . S i n c e we a r e i n t e r e s t e d i n h a v i n g a u n i q u e p o i n t o f e q u i l i b r i u m we w i l l h a v e t o i m p o s e a n e x t r a c o n d i t i o n : C o n d i t i o n 1. T h e c u r v e F ( x , y ) = o i s s i t u a t e d b e t w e e n t h e x - a x i s a n d t h e s t r a i g h t l i n e f ( f f * ) = o . O b v i o u s l y c o n d i t i o n 1 i m p l i e s t h e c o n d i t i o n (5.17). I t i s 51 not d i f f i c u l t to see that — = F(x,y) > o for the points lying to the right of the curve F(x,y) = o If = F(x,y) < o for the points lying to the l e f t of the curve F(x,y) = o = f(«0 >o for the points below the straight line f (c) = o = f ( •*) Co for the points above the straight line f (<r») = o The function y(t) i s maximum for y > o and minimum for y < o for the points on the straight line f(«*) = o and x(t) at tains maximum for x > o and minimum for x < o for the points lying on the curve F(x,y) = o, The direction, of motion i s shown in f i g . 9 . The curve F(x,y) = o, the straight line f (-*>) = o and the axes of co-ordinates divide the plane (x,y) into eight re gions. As before we introduce polar co-ordinates x = r cos 4, , y = r sin <t>. Then r = x cos <P + y s i n * r«j> = -x sin 4» + y cos«fr F I G . 9 The signs of r and <P in different regions are given as (1,5) r > o, *p may be Sfc. o (2,6) r may be ^  o, «* > o (3,7) r < o, 4> may be \ o (4,8) r may be \ o , <*> > o. 52 I n t h e r e g i o n s ( l ) a n d (5) we s e e t h a t r > o a n d «*» m a y b e \ o . F i r s t o f a l l we n e e d a c o n d i t i o n t h a t m a k e s 4> > o i n t h e s e r e g i o n s . T h i s i s n e c e s s a r y i n o r d e r t o e n s u r e t h a t t h e r e i s n o m o t i o n w i t h b o u n d e d p o l a r a n g l e i n t h e r e g i o n s ( l , 5 ) . O u r p u r  p o s e i s s e r v e d i f f o r e x a m p l e we a s s u m e t h a t % ( x , y ) < o , «f> ( x , y ) > o i n t h e r e g i o n ( l ) (5 . 1 8 ) - ^ ( x j y ) > o , «v ( x , y ) < 0 i n t h e r e g i o n (5) N e x t we r e q u i r e t h a t t h e r e b e n o p e r i o d i c m o t i o n s a n d f o r t h a t we m u s t h a v e - H - a< ¥ < ° ( 5 - 1 9 ) a c c o r d i n g t o B e n d i x s o n c r i t e r i o n . L e t a m o t i o n M . ( t ) e n t e r t h e r e g i o n ( 8 ) a f t e r i n t e r s e c t i n g t h e n e g a t i v e h a l f y - a x i s . I t i s e a s y t o s e e a s i n c a s e 2 t h a t t h e m o t i o n M ( t ) c r o s s e s t h e x - a x i s a n d e n t e r s t h e r e g i o n ( l ) . We n o w s h o w t h a t i t c a n n o t r e m a i n i n t h e r e g i o n ( l ) a n d m u s t e n t e r t h e r e  g i o n (2). T h i s i s d o n e a s f o l l o w s . We c o n s i d e r t h e r a t e o f c h a n g e o f ( c x - a y ) a l o n g t h e m o t i o n M ( t ) , i . e . , ( c x - a y ) = a c x - b e y + c *>i (x ,y) - a c x + a d y - a«s»(x,y) = y ( a d - b e ) + c > ( x , y ) - a ^ ( x , y ) < o (5.20) i n t h e r e g i o n ( l ) i n v i e w o f (5 . 1 7 ) a n d (5. 1 8 ) . We c o n s i d e r t h e s t r a i g h t l i n e s c x - a y = A ( A p o s i t i v e ) a n d c x - d y = o a n d f i n d t h e i r p o i n t o f i n t e r s e c t i o n (— _1_ , A \ \ c d - a d ^ a j I t l i e s i n t h e f i r s t q u a d r a n t s i n c e A , c , d , d - a a r e a l l p o s i  t i v e q u a n t i t i e s . S i n c e F ( x , y ) = o l i e s a l w a y s b e - l o w t h e s t r a i g h t 53 l i n e f (••) = o , t h e s t r a i g h t l i n e c x - a y = A i n t e r s e c t s t h e c u r v e F ( x , y ) = o f o r a l l A . T h u s t h e m o t i o n M ( t ) e n t e r i n g t h e r e g i o n (l) m u s t c r o s s t h e c u r v e F ( x , y ) = o , b e c a u s e i t c a n n o t g o t o t h e o r i  g i n s i n c e <p > o a n d c a n n o t c r o s s t h e s t r a i g h t l i n e c x - a y = A d u e t o (5.20). A f t e r i n t e r s e c t i n g t h e c u r v e F ( x , y ) = o i t c a n n o t e v e n r e m a i n i n t h e r e g i o n (2) b e c a u s e o f t h e s a m e r e a s o n . T h u s t h e m o t i o n e n t e r s t h e r e g i o n (3) w h e r e i t e i t h e r t e n d s t o t h e o r i g i n w i t h t h e i n c r e a s e o f t i m e o r g o e s o u t o f t h i s r e g i o n a n d e n t e r s t h e r e g i o n (4). S i m i l a r a r g u m e n t h o l d s f o r t h e r e g i o n s (4), (5), (6) a n d (7). T h i s s h o w s t h a t t h e r e c a n n o t b e a n y m o t i o n w i t h b o u n  d e d p o l a r a n g l e i n t h e r e g i o n s ( l ) , (2), (4), (5), (6) a n d (8). T h e m o t i o n w i t h b o u n d e d p o l a r a n g l e c a n o n l y o c c u r i n r e g i o n s (3) a n d (7) , w h e r e i t i s b o u n d e d , s i n c e r <: o . T h u s a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d . F o r t h e s t r a i g h t l i n e L(o,o») we c a n t a k e t h e p o s i t i v e h a l f x - a x i s . H e n c e a l l t h e r e q u i r e m e n t o f T h e o r e m 1.2.7 a r e s a t i s f i e d a n d we h a v e t h e f o l l o w i n g t h e o r e m : T h e o r e m 5.3. L e t _1Z > o . T h e n u n d e r c o n d i t i o n s (5.2) - (5.5), 2_X ; (5.16), (5.18). (5.19) a n d C o n d i t i o n 1, t h e t r i v i a l s o l u t i o n x=o=y  o f s y s t e m (5.l) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . I n 1954 G u , C a o h a o £l6] d i s c u s s e d t h e s t a b i l i t y o f t h e t r i v i a l s o l u t i o n i n t h e l a r g e o f t h e f o l l o w i n g s y s t e m § = x h ^ y ) + * ( y ) § = x h 2 ( y ) + f ( y ) b y c o n s t r u c t i n g a L y a p u n o v f u n c t i o n . I n t h e n e x t t w o s e c t i o n s we-54 s h a l l d i s c u s s t h e s t a b i l i t y i n t h e l a r g e o f i f = h i ( y ) x + a y J f = h g ( x ) x + b y a n d J f = x h ^ y ) + a y J f = b x + h 2 ( x ) y m o s t l y b y q u a l i t a t i v e m e t h o d s . 55 6. T h e s t a b i l i t y i n t h e l a r g e o f • d x - = h , ( y ) x + a y , d t 1 = h ( x ) x + b y . d t ^ L e t u s c o n s i d e r t h e s y s t e m H = ^ ( y ) x + a y J f = h 2 ( x ) x + b y (6.1) We a s s u m e t h a t t h e r i g h t h a n d s i d e s o f t h e s y s t e m ( 6 . l ) s a t i s f y t h e c o n d i t i o n s g u a r a n t e e i n g t h e e x i s t e n c e a n d u n i q u e n e s s o f e v e r y s o l u t i o n . 6a. L e t u s a s s u m e t h a t a h 2 ( x ) < o , x * o , b < o a n d ( y ) < o , y * o P o r t h e s a k e o f d e f i n i t e n e s s we t a k e a < o , h 2 ( x ) > o , x * o . We r e p r e s e n t t h e r i g h t h a n d s i d e s o f (6.1) o n t h e ( x , y ) p l a n e . T h e o r - d i n a t e s o f t h e g r a p h s o f t h e c u r v e s a r e g i v e n b y h i ( y i ) y-j - ~ ' — x >± o a c c o r d i n g a s x $ o h ( x ) ( 6 ' 2 ) ^2 ~ - ' — x >^o a c c o r d i n g a s x ^ o T h i s s h o w s t h a t t h e c u r v e s h ^ ( y ) x + a y = o a n d h 2 ( x ) x + b y = o l i e i n t h e s e c o n d , f o u r t h a n d f i r s t , t h i r d q u a d r a n t s r e s p e c t i v e l y a n d c o n s e q u e n t l y t h e o r i g i n i s t h e o n l y p o i n t o f e q u i l i b r i u m . I t i s n o t d i f f i c u l t t o s e e t h a t d x d t clx — = h ^ ( y ) x + a y > o t o t h e l e f t o f t h e c u r v e h ^ ( y ) x + a y = o = h ^ ( y ) x + a y < o t o t h e r i g h t o f t h e c u r v e h ^ ( y ) x + a y = o 56 | f = h 2 ( x ) x + b y < o t o t h e l e f t o f t h e c u r v e h^(x)x + b y = o | f = h 2 ( x ) x + b y > o t o t h e r i g h t o f t h e c u r v e h 2 ( x ) x + b y = o T h e f u n c t i o n y ( t ) a t t a i n s m a x i m u m o n t h e c u r v e h 2 ( x ) x + b y = o f o r y > o a n d m i n i m u m f o r y < o ; x ( t ) i s m i n i m u m o n t h e c u r v e h ^ ( y ) x + a y = o f o r y > o a n d m a x i m u m f o r y < o . T h e d i r e c t i o n o f m o t i o n a n d t h e c u r v e s o b t a i n e d b y p u t t i n g t h e r i g h t h a n d s i d e s o f ( 6 . l ) e q u a l t o z e r o a r e r e p r e s e n t e d i n f i g . 1 0 . T h e c u r v e s a n d t h e c o - o r d i n a t e a x e s ' d i v i d e ci t h e p l a n e i n t o e i g h t r e g i o n s . U s i n g p o l a r c o - o r d i n a t e s w e s e e t h a t t h e s i g n s o f r a n d ^» i n t h e s e r e g i o n s a r e (1.5) r m a y b e >^ o, tp> o (2.6) r < o , <$> m a y b e Tfa o (3.7) r may b e * t ^ o , . <p > o (4.8) r <• o , <p m a y b e o T h e t r i v i a l s o l u t i o n i s F I G . J . O e a s i l y s h o w n t o b e a s y m p t o t i c a l l y s t a b l e i n t h e s e n s e o f L y a p u n o v . I n f a c t , l e t a V - f u n c t i o n b e d e f i n e d b}J 2 V = 2 J x h 2 ( x ) d x - a y 2 o T h e n V = x h 2 ( x ) ( h 1 ( y ) x + a y ) - a y ( x h g ( x ) + b y ) 2 2 = x ( y ) h 2 ( x ) • a b y < o y * o = o p o s s i b l y f o r y = o 57 O b v i o u s l y , V i s a p o s i t i v e d e f i n i t e f u n c t i o n . T h e d e r i v a t i v e V i s n e g a t i v e f o r y * o a n d p o s s i b l y z e r o f o r y = o . S i n c e y = o d o e s n o t c o n t a i n a p o s i t i v e h a l f - t r a j e c t o r y o f t h e s y s t e m (6.1) e x c e p t x = o = y , t h e t r i v i a l s o l u t i o n i s a s y m p t o t i c a l l y s t a b l e a c c o r d i n g t o L y a p u n o v . S i n c e b + h ^ ( y ) < o , t h e r e a r e n o p e r i o d i c s o l u t i o n s a c c o r d i n g t o t h e c r i t e r i o n o f B e n d i x s o n . T h i s i s o b v i o u s f r o m t h e f a c t t h a t f o r t h e s y s t e m i s c o n s t r u c t e d a L y a p u n o v f u n c t i o n . P o r t h e s t r a i g h t l i n e L ( o , <*>) a p p e a r i n g i n T h e o r e m 1.2.7 we t a k e t h e p o s i t i v e h a l f y - a x i s . We n o w s h o w t h a t t h e m o t i o n s w i t h b o u n d e d p o l a r a n g l e s a r e b o u n d e d . We c o n s i d e r f i r s t t h e r e g i o n s (2,4,6,8). I n t h e s e r e g i o n s a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d , s i n c e r < o . N e x t we c o n s i d e r t h e r e g i o n s (l,3>5>7). H e r e w e s h o w t h a t t h e r e a r e n o m o t i o n s w i t h b o u n d e d p o l a r a n g l e s , i . e . , a n y m o t i o n s t a r t i n g o r e n t e r i n g t h e s e r e  g i o n s m u s t l e a v e t h e m w i t h t h e i n c r e a s e o f t i m e . P o r t h i s we w r i t e 4> = — [~ s i n * (ni W x + a y ) + c o s < * ( h 2 ( x ) x + b y ) L e t u s s u p p o s e t h a t t h e m o t i o n s t a r - t e d i n t h e r e g i o n ( l ) d o e s n o t c r o s s t h e c u r v e h ^ ( x ) x + b y = o t h e n , s i n c e x i s d e c r e a s i n g a n d y i s i n c r e a s i n g i n ( l ) , y b e c o m e s i n f i n i t e l y l a r g e a n d c o n s e  q u e n t l y > € > o a n d h e n c e w i t h t h e i n c r e a s e o f t i m e t h e m o t i o n l e a v e s t h e r e g i o n ( l ) . T h i s c o n t r a d i c t s o u r a s s u m p t i o n . T h e c o n t r a  d i c t i o n s h o w s t h a t t h e m o t i o n m u s t l e a v e t h e r e g i o n ( l ) a n d e n t e r t h e r e g i o n (2). T h e same r e a s o n i n g h o l d s f o r t h e r e g i o n (5). N e x t we c o n s i d e r t h e r a t e o f c h a n g e o f t h e q u a n t i t y ( b x - a y ) a l o n g t h e t r a j e c t o r i e s o f ( 6 . l ) 58 — ( b x - a y ) = b ( ^ ( y ) x + a y ) - a ( h 2 ( x ) x + b y ) (6.3) = x ( b h ^ ( y ) - a h 2 ( x ^ o a c c o r d i n g a s x ^ o C o n s i d e r t h e s t r a i g h t l i n e b x - a y = A (6.4) T h e s t r a i g h t l i n e (6.4) i n t e r s e c t s t h e c u r v e x h ^ ( y ) + a y = o f o r a l l A . L e t A b e n e g a t i v e , t h e n t h e s t r a i g h t l i n e (6.4) i n t e r s e c t s t h e c u r v e h ^ ( y ) x + a y = o i n . t h e f o u r t h q u a d r a n t . We c o n s i d e r t h e r e g i o n b o u n d e d b y t h e n e g a t i v e h a l f y - a x i s , t h e s t r a i g h t l i n e b x - a y = A a n d t h e c u r v e h ^ ( y ) x + a y = o . T h e m o t i o n e n t e r i n g t h i s r e g i o n m u s t c r o s s t h e c u r v e h ^ ( y ) x + a y = o , s i n c e i t c a n n o t c r o s s t h e s t r a i g h t l i n e b x - a y = A b e c a u s e o f (6.3) a n d c a n n o t e n t e r t h e o r i g i n , s i n c e <p > o i n t h i s r e g i o n . S i m i l a r a r g u m e n t h o l d s i n r e  g i o n (3). T h e a b o v e a n a l y s i s s h o w s t h a t a l l m o t i o n s w i t h b o u n d e d p o l a r a n g l e s a r e b o u n d e d a n d we h a v e t h e f o l l o w i n g t h e o r e m ; T h e o r e m 6.1. I f a h 2 ( x ) < o f o r x $ o , b < o a n d h ^ ( y ) < o f o r y * o , t h e n t h e t r i v i a l s o l u t i o n o f ( 6 . l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e  l a r g e . 6b. We s h a l l n o w d i s c u s s t h e c a s e w h e n a h 2 ( x ) > o a n d t h e c o n d i  t i o n s b + ( y ) < o , bh. j ( y ) - a h 2 ( x ) > o , x * o , y $ o (6.5) a r e s a t i s f i e d . F r o m (6.5) i t f o l l o w s t h a t i f a h 2 ( x ) > o we m u s t n e c e s s a r i l y h a v e b < o , h ^ ( y ) < o f o r y # o . We a s s u m e a > o , h ? ( x ) > o , x t o . F r o m (6.2) we h a v e 59 \ ( y 1 ) h g ( x ) y i - y ? = " x ~ i + x b b h ( ) - a h M ( 6 > 6 ) _ _ x ^ ^ 2 ^ x ^ > o a c c o r d i n g a s x >^o a b S i n c e a n d y 2 a r e p o s i t i v e o r n e g a t i v e a c c o r d i n g a s x i s p o s i t i v e o r n e g a t i v e , t h e t w o c u r v e s x h 2 ( x ) + b y = o (6.7) x h 1 ( y ) + a y = o a r e s i t u a t e d i n t h e f i r s t a n d t h i r d q u a d r a n t s a n d s u c h t h a t t h e c u r v e x h ^ ( y ) + a y = o i s a b o v e t h e c u r v e x h ^ ( x ) + b y = o i n t h e f i r s t q u a  d r a n t a n d b e l o w i n t h e t h i r d q u a d r a n t ( b e c a u s e o f (6.6)). We f u r t h e r n o t e t h a t d x d t d x d t = b ^ ( y ) x + a y •> o t o t h e l e f t o f t h e c u r v e x h ^ ( y ) + a y = o = h ^ ( y ) x + a y <o t o t h e r i g h t o f t h e c u r v e x h ^ ( y ) + a y = o |f = h ^ ( x ) x + b y <o t o t h e l e f t o f t h e c u r v e x h 2 ( x ) + b y = o |f = h 2 ( x ) x + b y >o t o t h e r i g h t o f t h e c u r v e x h 2 ( x ) + b y = o T h e f u n c t i o n x ( t ) i s m a x i m u m o n t h e c u r v e x h ^ ( y ) + a y = o f o r y > o a n d m i n i m u m f o r y 4 o; y ( t ) i s m a x i m u m o n t h e c u r v e x h 2 ( x ) + b y = o f o r y > o a n d m i n i m u m f o r y < o . T h e c u r v e s a n d t h e c o - o r d i n a t e a x e s d i v i d e t h e fc»y) p l a n e i n t o e i g h t r e g i o n s . T h e d i r  e c t i o n o f m o t i o n a n d t h e c u r v e s a r e s h o w n i n f i g . 1 1 . A s b e f o r e we i n t r o d u c e p o l a r c o - o r d i n a t e s a n d n o t i c e t h a t t h e s i g n s o f r a n d e*» i n d i f f e r e n t r e g i o n s a r e a s f o l l o w s ; (1,5) r may b e ^ o , <i> < o (2,6) f < o , <p m a y b e ^ o 60 o (3.7) r m a y b e \ o , 4> > o (4.8) r < o , 4> m a y b e ^ o I n e a c h o f t h e r e g i o n s (4) a n d (8) t h e r e w i l l b e a t l e a s t o n e i n  t e g r a l c u r v e g o i n g t o t h e o r i g i n , t h e o n l y p o i n t o f e q u i l i b r i u m . T h i s f o l  l o w s f r o m T h e o r e m 2.1 o f E r u g i n ' s w o r k f 8 j . O t h e r m o t i o n s s t a r t e d i n t h e r e  g i o n s (4) a n d (8) e i t h e r g o t o t h e o r i g i n o r e n t e r t h e r e g i o n s (3,5) o r (1,7) F I G . i l ( s i n c e r < o , <*> m a y b e ^ o i n t h e r e g i o n s (4,8)). We n o w s h o w t h a t m o t i o n s e n t e r i n g t h e r e g i o n s ( l ) , (3), (5) a n d (7) m u s t l e a v e t h e s e r e g i o n s a n d e n t e r t h e r e g i o n s (2) o r (6). T o s h o w t h i s we c o n s i d e r t h e r a t e o f c h a n g e o f t h e q u a n t i t y ( b x - a y ) , i . e . j d — ( b x - a y ) = x ( b h ^ ( y ) - a l v ^ x ) ) ^ o a c c o r d i n g a s x \ o (6.8) T h e s t r a i g h t l i n e b x - a y = A ( A n e g a t i v e ) h a s p o s i t i v e i n t e r c e p t s w i t h t h e a x e s o f c o - o r d i n a t e s a n d h e n c e i n t e r s e c t s t h e c u r v e s (6.7) i n t h e f i r s t q u a d r a n t . C o n s i d e r t h e r e g i o n b o u n d e d b y t h e s t r a i g h t l i n e b x - a y = A ( - v e ) , t h e c u r v e x h ^ ( y ) + a y = o a n d t h e p o s i t i v e h a l f y - a x i s . T h e m o t i o n M ( t ) e n t e r i n g t h i s r e g i o n c a n n o t c r o s s t h e l i n e b x - a y = A ( A n e g a t i v e ) b e c a u s e o f (6.8) a n d c a n n o t g o t o t h e o r i g i n , s i n c e < o a n d h e n c e m u s t l e a v e t h i s r e g i o n w i t h t h e i n c r e a s e o f t i m e . S i m i l a r a r g u m e n t h o l d s f o r t h e r e g i o n s (3), (5) a n d (7). T h e m o t i o n s a f t e r e n t e r i n g t h e r e g i o n s (2) o r (6) t e n d t o t h e o r i g i n a s t _ > 00 . T h u s we h a v e p r o v e d t h e f o l l o w i n g t h e o r e m : 61 T h e o r e m 6.2, I f a h 2 ( x ) > o f o r x $ o , t h e n t h e t r i v i a l s o l u t i o n o f ( 6 , l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r c o n d i t i o n s (6 ,5 ) . 6c. We n o w a s s u m e t h a t e i t h e r b > o o r h ^ ( y ) > o f o r y + o a n d t h e c o n d i t i o n s (6 .5) a r e s a t i s f i e d . P o r t h e s a k e o f d e f i n i t e n e s s we l e t b > o . T h e n f r o m (6.5) i t f o l l o w s t h a t h ^ ( y ) < o a n d a h g ( x ) < o . We l e t a < o , h 2 ( x ) > o f o r x $ o . T h e o r d i n a t e s o f t h e c u r v e s i n (6.7) a r e y* = - x . 1 ^ ^ o a c c o r d i n g a s x ^ o a h , ( x ) y 9 = - x _ & ^ o a c c o r d i n g a s x £ o We c o m p u t e t h e d i f f e r e n c e b e t w e e n t h e o r d i n a t e s h ( y ) h ( x ) y . - y ? = - x + x -Ar 1 2 a b (6.10) b h ( y ( ) - a h p ( x ) — 3 — ' 6 > o a c c o r d i n g a s x ^ o = - x a b P r o m (6.9) a n d (6.10) i t f o l l o w s t h a t t h e t w o c u r v e s l i e i n t h e s e  c o n d a n d f o u r t h q u a d r a n t s a n d t h e c u r v e x h ^ ( y ) + a y = o l i e s a b o v e t h e c u r v e x h g ( x ) + b y = o i n t h e f o u r t h q u a d r a n t a n d b e l o w i n t h e s e c o n d q u a d r a n t . I t i s n o t d i f f i c u l t t o v e r i f y t h a t d x d t | f = h ^ ( y ) x + a y < o t o t h e r i g h t o f t h e c u r v e h ^ ( y ) x + a y = | f = h 2 ( x ) x + b y > o t o t h e r i g h t o f t h e c u r v e h 2 ( x ) x + b y = | f = h 2 ( x ) x + b y <. o t o t h e l e f t o f t h e c u r v e h 2 ( x ) x + b y = o T h e f u n c ' t i o n x ( t ) i s m i n i m u m o n t h e c u r v e x h ^ ( y ) + a y = o = h.| ( y ) x + a y > o t o t h e l e f t o f t h e c u r v e h ^ ( y ) x + a y = o 62 f o r y > o a n d m a x i m u m f o r y < o , a n d y ( t ) i s m a x i m u m o n t h e c u r v e x h 2 ( x ) + b y = o f o r y > o a n d m i n i m u m f o r y < o . T h e c u r v e s (6.7) a n d t h e c o - o r d i n a t e a x e s d i v i d e t h e p l a n e i n t o e i g h t r e g i o n s . T h e c u r v e s a n d t h e d i r e c t i o n o f m o t i o n a r e s h o w n i n f i g . 1 2 . T h e s i g n s o f r a n d 4> i n d i f f e r e n t r e g i o n s a r e (1.5) r m a y b e \ o, <*> > o (2.6) r > o , ep m a y b e ^ o (3.7) r m a y b e >^o, <p > o (4.8) r < o , may b e ^ 0 L e t u s c o m p u t e 4> i n t h e r e  g i o n s (2) a n d (6) a n d s e e w h e t h e r i t i s p o s i t i v e o r n o t . L e t c,| b e t h e g r e a t e s t l o w e r b o u n d o f t h e f u n c t i o n h ^ y ) . F IG. 1 2 1 i s f i n i t e o t h e r w i s e b h ^ ( y ) - a f c ^ x ) > o f o r x t o , y + o i s v i o l a t e d . We t h e n h a v e c^ ^ h ^ ( y ) < - b F r o m t h e i n e q u a l i t y b h ^ ( y ) - a h 2 ( x ) > o we h a v e b c . j - a h 2 ( x ) o . T h e e q u a l i t y s i g n i s a d m i t t e d i f h ^ ( y ) d o e s n o t a t t a i n i t s g r e a t e s t l o w e r b o u n d o t h e r w i s e s t r i c t i n e q u a l i t y h o l d s . L e t u s s u p p o s e t h a t c^ i s a t t a i n t e d , t h e n c i $ Vy) < - b (6.11) a n d h „ ( x ) > bcT, * a I f we l e t ^ ( y ) = + * ( y ) , t h e n o ( y ) < - ( b + c 1 ) 63 f r o m (6.1l). P o r h 2 ( x ) we c a n t a k e h 2 ( x ) = C 1 + » ^ ( x ) , w h e r e ^ ( x ) > o 3. N o w • • • r <*• = - x s i n * + y c o s * = - s i n <p ( x h ^ y ) + a y ) + c o s <p ( h 2 ( x ) x + b y ) = - sin4>J^x(c1 + ^ ( y ) ) + a y j + C o s * ^ ( ^ | i + ^ ( x ) ) x + b y ] T h e n 2 b e . 2 <P = - a s i n * + 1 c o s <p + (b - c, ) s i n * c o s * a 1 2 - ^ ( y ) s i n 4. c o s * + c o s * 2 f , 2 b - c , b c i 1 /• / v = - a c o s * I t a n <p - — _ L t a n 4» - —rr- 1 - ( y j s i n * c o s <$> I a a ^ J 1 2 + £ p ( x ) c o s * ( * * m u l t i p l e o f ^ ) 2 2 ID c = - a c o s < * ( t a n <P - 3;) ( t a n 4> + .21) - & ( y ) s i n 4> c o s 4> + ^ 2(x) c o s 2 * (6.12) T h e e x p r e s s i o n ( t a n * - ^ ( t a n ^ + c 1 ) c a n c h a n g e i t s a ~ • b C 1 s i g n o n l y w h i l e p a s s i n g t h r o u g h t h e v a l u e s — a n d - — — T h e v a l u e & a. — > - —L s i n c e b + c . < o a n d a < o . S i n c e — > ~ c 1 , t h e s t r a i g h t a a ' a a 7 l i n e w i t h s l o p e — l i e s a b o v e t h e s t r a i g h t l i n e w i t h s l o p e - 2l. f o r a a x > o a n d b e l o w f o r x < o . T h e s t r a i g h t l i n e s y = - r x a n d y = - °2_ x & 2L l i e i n t h e s e c o n d a n d f o u r t h q u a d r a n t s . We s h o w t h a t t h e y d o n o t l i e i n t h e r e g i o n s ( 2 ) a n d (6). P o r t h i s w e h a v e o n l y t o s h o w t h a t t h e s t r a i g h t l i n e y = - ^ 1 _ x l i e s b e l o w t h e c u r v e x h _ ( x ) + b y = o a ^ 64 i n t h e s e c o n d q u a d r a n t a n d a b o v e i n t h e f o u r t h q u a d r a n t . I n f a c t > t h i s i s s o , b e c a u s e h2^ x) - c1 x/ b c1 - x — r — + — x = r - ( - h ? ( x ) ) >± o a c c o r d i n g a s x £ o T h u s we s e e f r o m (6.12) t h a t <* k e e p s t h e s a m e s i g n t h r o u g h o u t t h e r e g i o n s (2) a n d (6) a n d w h i c h i s e a s i l y s e e n t o b e p o s i t i v e . A c c o r d i n g t o c o n d i t i o n s (6.5)>the o r i g i n i s t h e o n l y p o i n t o f e q u i l i b r i u m a n d t h e r e a r e n o p e r i o d i c m o t i o n s . L e t a m o t i o n M ( t ) s t a r t i n r e g i o n ( l ) . T h e m o t i o n ;,M(fc)must e n t e r t h e r e g i o n (2) w i t h t h e i n c r e a s e o f t i m e , o t h e r w i s e y b e c o m e s i n f i n i t e a n d t h e n f r o m — = h j ( y ) x + a y i t f o l l o w s t h a t t h e m o t i o n M ( t ) c a n n o t r e m a i n i n t h e f i r s t q u a d r a n t . We n o w s h o w t h a t i t c a n n o t r e m a i n i n t h e r e g i o n (2) f o r a l l t i m e . F o r t h i s we c o n s i d e r t h e s t r a i g h t l i n e s b x - a y = A a n d a y + c ^ x = o / A C 1 A \ T h e i r p o i n t o f i n t e r s e c t i o n i s g i v e n b y ^ ^ j 1 , - a ( b + e )J * T h i s l 3 - e s i n t h e s e c o n d o r f o u r t h q u a d r a n t a c c o r d i n g a s A i s p o s i t i v e o r n e g a  t i v e . We c o n s i d e r t h e r e g i o n b o u n d e d b y t h e s t r a i g h t l i n e b x - a y = A ( + v e ) , t h e p o s i t i v e h a l f y - a x i s a n d t h e s t r a i g h t l i n e a y + c ^ x = o . S i n c e — ( b x - a y ) < o i n t h e s e c o n d q u a d r a n t a n d <p > o i n (2), i t f o l  l o w s t h a t t h e m o t i o n e n t e r s t h e r e g i o n (3). H e r e , i n t h e r e g i o n (3)> s i n c e • > o a n d t h e c u r v e x h ^ ( y ) + a y = o l i e s a b o v e t h e s t r a i g h t l i n e b x - a y = q a n d y <. o , t h e m o t i o n e n t e r s t h e r e g i o n (4) w i t h i n c r e a s e o f t i m e . T h e m o t i o n a f t e r e n t e r i n g t h e r e g i o n (4) e i t h e r t e n d s t o t h e o r i g i n o r e n t e r s t h e r e g i o n ( 3 ) , s i n c e r < o i n r e g i o n (4). S i m i l a r r e a s o n i n g s h o l d f o r t h e r e g i o n s (5), (6), (7) a n d (8). T h e a b o v e a n a l y  s i s s h o w s t h a t a n y m o t i o n w i t h b o u n d e d p o l a r a n g l e i s b o u n d e d . F o r t h e 65 s t r a i g h t l i n e L ( o , a * ) a p p e a r i n g i n T h e o r e m 1 . 2 . 7 we c a n t a k e p o s i  t i v e h a l f y - a x i s . Now i f we a s s u m e t h a t t h e t r i v i a l s o l u t i o n i s a s y m p  t o t i c a l l y s t a b l e i n t h e s e n s e o f L y a p u n o v , t h e n we h a v e p r o v e d t h e f o l l o w i n g t h e o r e m . * T h e o r e m 6.3. I f e i t h e r b > o o r h ^ ( y ) > o , y $ o a n d c o n d i t i o n s (6.5) a r e s a t i s f i e d , t h e n t h e t r i v i a l s o l u t i o n o f s y s t e m ( o . l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e p r o v i d e d i t i s a s y m p t o t i c a l l y s t a b l e i n t h e s m a l l ( i . e . , a c c o r d i n g t o L y a p u n o v ) . T h e r e q u i r e m e n t t h a t t h e t r i v i a l s o l u t i o n b e s t a b l e i n t h e s m a l l c a n b e r e a l i z e d , i f we t a k e , f o r e x a m p l e , h p ( x ) = — — + * ^ o ( x ) = — — + m + £0(x), w h e r e m > o a n d ^Ax] & a. <- d o e s n o t c o n t a i n a n y c o n s t a n t t e r m a n d h ^ ( y ) = c + * ^ ( y ) , w h e r e ^ ( y ) d o e s n o t c o n t a i n a n y c o n s t a n t t e r m . T h e e q u a t i o n s o f f i r s t a p p r o x i m a  t i o n c a n b e w r i t t e n a s d x at = ° 1 X + ^ T h e c h a r a c t e r i s t i c e q u a t i o n i s ?» - Oj a bc_j + m > - b a = o o r ^ - ( b + c . j ) A - a m = o T h e r o o t s o f t h i s e q u a t i o n h a v e n e g a t i v e r e a l p a r t s 3 s i n c e b + c„ < o a n d - am > o . w h i c h e n s u r e s t h e s t a b i l i t y i n t h e s m a l l a n d w h i c h i n t u r n , e n s u r e s 66 t h e s t a b i l i t y i n t h e l a r g e . T h e o r i g i n i s a n o d e o r a f o c u s a c c o r d i n g a s A = ( b + c . ) + 4 a m ^ o . I n c a s e A > o , t h e d i r e c t i o n s a l o n g w h i c h t h e m o t i o n s t e n d t o t h e o r i g i n a r e g i v e n b y a u + ( a , - b ) u •1 + m) = o T h i s i s o b t a i n e d b y p u t t i n g <#» = o i n t h e e x p r e s s i o n - r 4» = s i n 4> (c^x + a y ) - c o s • + m) x + b y ) a n d b y w r i t i n g u = t a n * , w h e r e t a n * = •L-. T h e t w o d i r e c t i o n s a r e c a l l e d c r i t i c a l d i r e c t i o n s . A c r i t i c a l d i r e c t i o n i s c a l l e d s i n g u l a r i f i t s a t i s f i e s t h e e q u a t i o n o t h e r w i s e i t i s a n o r d i n a r y c r i t i c a l d i r e c t i o n . A l o n g t h e o r d i n a r y c r i t  i c a l d i r e c t i o n e i t h e r e n t e r s a n i n f i n i t e n u m b e r o f t r a j e c t o r i e s o r o n l y o n e i n t e g r a l c u r v e . T h e q u e s t i o n w h e t h e r a l o n g a p a r t i c u l a r o r d i n a r y d i r e c t i o n e n t e r s a f i n i t e n u m b e r o f t r a j e c t o r i e s o r o n l y o n e c a n b e d e c i d e d b y u s i n g F r o m m e r ' , c r i t e r i o n [*15j. T h e o r e m 6.4. I f b = o , t h e n t h e t r i v i a l s o l u t i o n o f ( 6 . l ) i s a s y m p t o   t i c a l l y s t a b l e i n t h e l a r g e u n d e r c o n d i t i o n s (6.5) c . + a u = o , O n s i m i l a r l i n e s i t i s e a s y t o p r o v e t h e f o l l o w i n g t h e o r e m ; 67 7. T h e s t a b i l i t y i n t h e l a r g e o f ^ = xh^Cy) + a y , J f = b x + h 2 ( x ) y . L e t u s c o n s i d e r t h e s y s t e m J f = ^ ( y ) + ay d v ( 7 - l } | f = b x + h 2 ( x ) y We a s s u m e t h a t t h e r i g h t h a n d s i d e s o f (7 . l ) s a t i s f y c o n d i - t i o n s w h i c h g u a r a n t e e t h e e x i s t e n c e a n d u n i q u e n e s s o f e v e r y s o l u t i o n . We f i r s t p r o v e t h e f o l l o w i n g t h e o r e m : T h e o r e m 7 . 1 . I f a b < o a n d h^Cy)^ o , y * o ; h 2 ( x ) $ o , x # o (at l e a s t i n o n e o f t h e s e c o n d i t i o n s t h e s t r i c t i n e q u a l i t y i s s a t i s f i e d ) , t h e n t h e t r i v i a l s o l u t i o n o f ( 7 . l ) i s a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e . P r o o f . We a s s u m e a < o , b > o . A L y a p u n o v f u n c t i o n f o r ( 7 . l ) u n d e r t h e c o n d i t i o n s o f t h e t h e o r e m i s 2 V ( x , y ) = b x 2 - a y 2 I t s t o t a l t i m e d e r i v a t i v e i n v i e w o f ( 7 . l ) i s V = b x ( x h 1 ( y ) + a y ) - a y ( b x + hg (x ) ' y ) 2 2 = b x ( y ) - a y h 2 ( x ) < o f o r x * o , y * o = o p o s s i b l y o n x = o o r y = o O b v i o u s l y , V ( x , y ) i s a n i n f i n i t e l y l a r g e p o s i t i v e d e f i n i t e f u n c t i o n a n d x = o o r y = o d o e s n o t c o n t a i n a n y p o s i t i v e h a l f t r a  j e c t o r y o f (7 . 1 ) e x c e p t t h e o r i g i n . H e n c e a l l t h e c o n d i t i o n s o f T h e o r e m 1 .2.5 a r e s a t i s f i e d w h i c h p r o v e s t h e a b o v e t h e o r e m . N e x t we c o n s i d e r t h e s y s t e m (7 . 1 ) u n d e r c o n d i t i o n s hL ,(y) + h 2 ( x ) < o , h ^ y ) hg(x) - a b > o f o r x * o y * o 68 T h e f o l l o w i n g t h e o r e m c a n b e p r o v e d i n t h e same w a y a s T h e o r e m 4.1. T h e o r e m 7.2. I f a b > o . t h e n t h e t r i v i a l s o l u t i o n o f (7.1) i s a s y m p   t o t i c a l l y s t a b l e i n t h e l a r g e u n d e r c o n d i t i o n s (7.2). T h e p r o o f o f T h e o r e m 7.3 g o e s o n s i m i l a r l i n e s a s t h e p r o o f o f T h e o r e m 6.3. T h e o r e m 7.3. I f e i t h e r h ^ ( y ) > o o r h ^ x ) > o a n d c o n d i t i o n s (7.2) a r e s a t i s f i e d , t h e n t h e t r i v i a l s o l u t i o n o f s y s t e m ( 7 . l ) i s a s y m p t o   t i c a l l y s t a b l e i n t h e l a r g e p r o v i d e d i t i s a s y m p t o t i c a l l y s t a b l e i n  t h e s m a l l . 69 8. R e m a r k s . T h e r e a r e some q u e s t i o n s r e m a i n i n g t o b e a n s w e r e d i n c o n n e c t i o n w i t h t h e r e s u l t s we h a v e o b t a i n e d . 8.1. T h e f i r s t q u e s t i o n a r i s e s i n c o n n e c t i o n w i t h T h e o r e m 2.2.1. We s h o w e d t h a t s t a b i l i t y i n t h e l a r g e h o l d s i f i n a d d i t i o n t o (2.2.2) - (2.2.5) e i t h e r c o n d i t i o n s A a n d B o r A a n d C a r e s a t i s  f i e d . I s i t n o t p o s s i b l e t o d e r i v e n e c e s s a r y a n d s u f f i c i e n t c o n  d i t i o n s ? I n t h e a u t h o r ' s v i e w i t i s m o s t u n l i k e l y . T h e s e c o n d q u e s  t i o n w h i c h t h e n n a t u r a l l y a r i s e s i s t h i s : w h a t s p e c i a l f o r m s h o u l d P ( x , y ) a n d f (o^ ) h a v e i n o r d e r t h a t t h e c o n d i t i o n s c o u l d b e n e c e s s a r y a s w e l l a s s u f f i c i e n t ? H a v i n g f o u n d t h i s , t h e p r o b l e m o f b o u n d a r i e s a n d r e g i o n s o f s t a b i l i t y c o u l d b e d i s c u s s e d i n t h o s e c a s e s w h e r e t h e s t a b i l i t y i n t h e l a r g e d o e s n o t h o l d . 8.2. We d i s c u s s e d t h e s t a b i l i t y o f t h e t r i v i a l s o l u t i o n o f s y s t e m s (6.1) a n d (7.1) u s i n g q u a l i t a t i v e m e t h o d s . We w e r e n o t a b l e t o c o n  s t r u c t L y a p u n o v f u n c t i o n s f o r t h e t w o s y s t e m s . I s i t p o s s i b l e t o c o n s t r u c t a L y a p u n o v f u n c t i o n f o r t h e s y s t e m | | = x h 1 ( y ) + a y |f = f ( x ) + h 2 ( x ) y o f w h i c h (6.1) a n d (7.1) a r e p a r t i c u l a r c a s e s u n d e r s u i t a b l e c o n  d i t i o n s o n t h e r i g h t h a n d s i d e s o f t h e a b o v e s y s t e m ? I t i s t h e a u t h o r ' s a i m t o i n v e s t i g a t e t h e s e q u e s t i o n s i n t h e f u t u r e . 70 B I B L I O G R A P H Y 1. A i z e r m a n , M . A . , O n a p r o b l e m c o n c e r n i n g t h e s t a b i l i t y i n t h e l a r g e o f d y n a m i c a l s y s t e m s , U s p e h i M a t . N a u k ( N . S . ) 4(1949), n o . 4 ( 2 8 ) a 187 - 1 8 8 . ( R u s s i a n ) 2. B a r b a s h i n , E . A . ; N . N . , K r a s o v s k i l , O n t h e s t a b i l i t y o f m o t i o n i n t h e l a r g e , D o l c l . A k a d . N a u k S S S R ( N . S . ) 86 0952), 453 - 456. ( R u s s i a n ) 3. B a r b a s h i n , E . A . , O n s t a b i l i t y o f n o n l i n e a r e q u a t i o n o f t h e t h i r d o r d e r , P r i k l . M a t . M e h . 16(1952), 629 - 632. ( R u s s i a n ) 4. B a r b a s h i n , E . A . ; N . N . , K r a s o v s k i l , O n t h e e x i s t e n c e o f L y a  p u n o v f u n c t i o n s i n t h e c a s e o f a s y m p t o t i c s t a b i l i t y i n t h e l a r g e , P r i k l . M a t . M e h . 18(1954), 345 - 350. ( R u s s i a n ) 5. C a r t w r i g h t , M . L . , O n s t a b i l i t y o f s o l u t i o n o f c e r t a i n d i f  f e r e n t i a l e q u a t i o n s o f t h e f o u r t h o r d e r , Q u a r t . J . o f M e c h . 9(1956), 185 - 193. 6. E r s h o v , B . A . , O n t h e s t a b i l i t y i n t h e l a r g e o f o u c e r t a i n s y s t e m o f a u t o m a t i c r e g u l a t i o n , P r i k l . M a t . M e h . 17(1953)> 61 - 72. ( R u s s i a n ) 7. E r s h o v , B . A . , One t h e o r e m o n s t a b i l i t y o f m o t i o n i n t h e l a r g e , P r i k l . M a t . M e h . 18(1954), 381 - 383. ( R u s s i a n ) 8. E r u g i n , N . P . , On c e r t a i n q u e s t i o n s a b o u t t h e s t a b i l i t y o f m o t i o n a n d q u a l i t a t i v e t h e o r y o f d i f f e r e n t i a l e q u a t i o n s i n t h e l a r g e , P r i k l . M a t . M e h . 14(1950), 459 - 512. ( R u s s i a n ) 9. E r u g i n , N . P . , Q u a l i t a t i v e i n v e s t i g a t i o n o f i n t e g r a l c u r v e s o f a s y s t e m o f d i f f e r e n t i a l e q u a t i o n s , P r i k l . M a t . M e h . 14(1950), 659 - 664. ( R u s s i a n ) 10. E r u g i n , N . P . , Some g e n e r a l q u e s t i o n s o f t h e t h e o r y o f s t a b i l i t y o f m o t i o n , P r i k l . M a t . M e h . 15(l95l), 2 2 8 - 236. ( R u s s i a n ) 11. E r u g i n , N . P . , On t h e c o n t i n u a t i o n o f s o l u t i o n s o f d i f f e r e n t i a l e q u a t i o n s , P r i k l . M a t . M e h . 15(l95l), 55 - 58. ( R u s s i a n ) 12. E r u g i n , N . P . . T h e o r e m o n i n s t a b i l i t y , P r i k l . M a t . M e h . 16(1952), 355 - 361. ( R u s s i a n ) 71 13. E r u g i n , N . P . , On o n e p r o b l e m o f t h e o r y o f s t a b i l i t y o f a s y s  t e m o f a u t o m a t i c r e g u l a t i o n , P r i k l . M a t . M e h . 16(1952), 620 - 628. ( R u s s i a n ) 14. E r u g i n , N . P . , M e t h o d s o f A . M . L y a p u n o v a n d q u e s t i o n s o f s t a  b i l i t y i n t h e l a r g e , P r i k l . M a t . M e h . 17(1953), 389 - 400. ( R u s s i a n ) 15. F r o m m e r , M . , D i e i n t e g r a l k u r v e n e i n e r g e w o h n l i e h e n B i f f e r e n t i a l - g l e i c h u n g e r s t e r O r d n u n g i n d e r U m f l e b u n g r a t i o n a l e r U n b e s t i m m t h e i t s s t e l l e n , M a t h . A n n . 99(1928), 222 - 272. 16. G u , G a o - h a o . O n s t a b i l i t y o f t w o e q u a t i o n s , A c t a . M a t h . S i n i c a 4(1954), 347 - 357. ( C h i n e s e w i t h R u s s i a n S u m m a r y ) 1 7 . K r a s o v s k i l , N . N . , O n s t a b i l i t y o f s o l u t i o n s o f o n e n o n l i n e a r s y s t e m o f t h r e e e q u a t i o n s i n t h e l a r g e , P r i k l . M a t . M e h . 17(1953), 339 - 350. ( R u s s i a n ) 1 8 . K r a s o v s k i l , N . N . , O n s t a b i l i t y o f s o l u t i o n s o f s y s t e m s o f t w o d i f f e r e n t i a l e q u a t i o n s , P r i k l . M a t . M e h . 17(1953), 651 - 672. ( R u s s i a n ) 19. K r a s o v s k i l , N . N . , O n s t a b i l i t y i n t h e l a r g e u n d e r c o n s t a n t l y a c t i n g d i s t u r b a n c e s , P r i k l . M a t . M e h . 18(1954), 95 - 102. ( R u s s i a n ) 20. K r a s o v s k i l , N . N . , O n t h e r e v e r s i b i l i t y o f t h e o r e m o f K . P . P e r s i d s k i i o n u n i f o r m s t a b i l i t y , P r i k l . M a t . M e h . 19(1955), 273 - 278. ( R u s s i a n ) 21. M a l k i n , I . G . , O n o n e p r o b l e m o f t h e o r y o f s t a b i l i t y o f a s y s  t e m o f a u t o m a t i c r e g u l a t i o n , P r i k l . M a t . M e h . 16(1952), 365 - 368. ( R u s s i a n ) iit«. fw« 22. M a l k i n , I . G . , 0 n A q u e s t i o n o f r e v e r s i b i l i t y o f ^ t h e o r e m o f L y a  p u n o v a b o u t a s y m p t o t i c s t a b i l i t y , P r i k l . M a t . M e h . 18(1954), 129 - 138. ( R u s s i a n ) 23. M a s s e r a , J . L . , O n L i a p o u n o f f ' s c o n d i t i o n s o f s t a b i l i t y , A n n . o f M a t h . (3) 50(1949), 705 - 721. 24. O g o r c o v , A . I . , O n s t a b i l i t y o f s o l u t i o n s o f t w o n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s o f t h e t h i r d a n d f o u r t h o r d e r s ; P r i k l . M a t . M e h . 23(1959), 179 - 1 8 1 . ( R u s s i a n ) 25. P l i s s , V . A . , Q u a l i t a t i v e p i c t u r e o f i n t e g r a l c u r v e s i n t h e l a r g e , P r i k l . M a t . M e h . 17(1953), 541 - 554. ( R u s s i a n ) 72 P l i s s , V . A . , N e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s o f s t a b i l i t y i n t h e l a r g e f o r a s y s t e m o f n - d i f f e r e n t i a l e q u a t i o n s , D o k l . A k a d . N a u k S S S R ( N . S . ) 103(1955), 17 - 1 8 . ( R u s s i a n ) P l i s s , V . A . , I n v e s t i g a t i o n 03 .a,; n o n - l i n e a r d i f f e r e n t i a l e q u a  t i o n o f t h e t h i r d o r d e r , D o k l . A k a d . N a u k S S S R ( N . S . ) 111(1956), 1178 - 1 1 8 0 . ( R u s s i a n ) P l i s s , V . A . , C e r t a i n p r o b l e m s o f t h e o r y o f s t a b i l i t y i n t h e l a r g e , L e n i n g r a d U n i v e r s i t y P u b l i s h i n g H o u s e 1958. ( R u s s i a n ) S i m a n o v , S . N . , O n s t a b i l i t y o f n o n l i n e a r e q u a t i o n o f t h e t h i r d o r d e r , P r i k l . M a t . M e n . 17(1953), 369 - 372. ( R u s s i a n ) W i n t n e r , A . , T h e n o n l o c a l e x i s t e n c e p r o b l e m o f o r d i n a r y d i f  f e r e n t i a l e q u a t i o n s , A m e r . J . M a t h . 67(1945), 277 - 284. Z u b o v , V . I . , p . Y o b X e T O s W f f i t t h e o r y o f ^ s e c o n d m e t h o d o f L y a p u n o v a n d c o n s t r u c t i o n o f g e n e r a l s o l u t i o n i n t h e r e g i o n o f a s y m p t o t i c s t a b i l i t y , P r i k l . M a t . M e h . 19(l955), 179 - 210. ( R u s s i a n ) 

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 5 1
Germany 3 2
China 2 28
Canada 1 0
Switzerland 1 0
City Views Downloads
Ashburn 5 0
Unknown 4 2
Shenzhen 2 28
Lausanne 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080607/manifest

Comment

Related Items