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Contribution to nonlinear differential equations Lalli, Bikkar Singh 1966

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A CONTRIBUTION TO NONLINEAR DIFFERENTIAL EQUATIONS by BIKKAR SINGH LALLI B.A. (Hons.), M.A. PANJAB U n i v e r s i t y , INDIA, 1951 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mathematics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1966 In presenting this thesis in pa r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely avail-able for reference and study. I further agree that permission-for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives, It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of H-^JU. v w ^ " e J i The University of B r i t i s h Columbia Vancouver 8, Canada THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE PINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY B. A. (Hons. ), M.A. Panjab U n i v e r s i t y , I n d i a , 1951 THURSDAY, August 11, 1966 at 3 : 3 0 p.m. Room 229, Mathematics B u i l d i n g COMMITTEE IN CHARGE Chairman: I. McT. Cowan E x t e r n a l Examiner: Dr. Joseph P. LaSal 1.-Dynamical Systems Research Centra Brown U n i v e r s i t y , Providence, Rhode i s l a n d o f BIKKAR SINGH LALLI D. E. B. Derry Leimanis N. Moyls C. W. C l a r k C. A. Swanson R. Westwick Research Supervisor: E. Leimanis A CONTRIBUTION TO NONLINEAR DIFFERENTIAL EQUATIONS Abstr a c t The subject matter of t h i s t h e s i s c o n s i s t s of a q u a l i t a t i v e study of the s t a b i l i t y and asymptotic s t a b i l i t y of the zero s o l u t i o n of c e r t a i n types of no n l i n e a r d i f f e r e n t i a l equations, f o r a r b i t r a r y i n i t i a l p e r t u r b a t i o n s , and the c o n s t r u c t i o n of a p e r i o d i c s o l u t i o n f o r a Hamiltonian system w i t h n(_> 2) degrees of freedom. In Chapter I the s t a b i l i t y of the auto-nomous system of two d i f f e r e n t i a l equations (1) x - x h 1 ( y ) + ay, y » f ( x ) + y h g ( x ) i s d i s c u s s e d , u s i n g q u a l i t a t i v e methods i n combination w i t h the c o n s t r u c t i o n of a Lyapunov f u n c t i o n . Various r e s u l t s proved by I. H. M u f t i become p a r t i c u l a r cases of our r e s u l t s . In the same chapter a g e n e r a l i z a t i o n of the problem of Aizerman f o r the case n = 2 i s given. In Chapter I I s t a b i l i t y of a Q u a s i l i n e a r equation (2) *x"+ a f 1 ( x , x ) x + f 2 ( x , x ) x + b f ^ ( x ) = 0 i s d i s c u s s e d , by u s i n g Lyapunov 1s second method. I t has been proved t h a t i f f-5 e C, i i v e t o x. ( i ) _ 5 f 2 € C 1 and f x € C 2 r e l a t -_ 2 2 ( i i ) f x ( x , y ) , g(x,y) > 1, - g ^ _< x f ^ < |-a,b > 0 f o r a l l values of x and y (x=y).. ( i i i ) ag(x,y) - b f ^ ( x ) > n 2 > 0 (n > 0) , f o r a l l x and y. ( i v ) y 3 g(x*y) < 0 and |x| (G(x,y) - l ) < x ^ 7 I I ( x , y ) | , y + 0. (v) W(x,y) - « as r = <Jx + y - c o where g,G, I and W are c e r t a i n f u n c t i o n s of f ^ , f g and f-j , then the zero s o l u t i o n of (2) 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 p e r t u r b a t i o n s . In the same chapter c e r t a i n equations of t h i r d order have a l s o been discussed f o r "complete s t a b i l i t y " . These equations are s p e c i a l cases of (2) and are more general than those con-s i d e r e d by Shimanov and Barbashin. Aizerman's problem f o r the case n = 3 i s g e n e r a l i z e d t o two d i f f e r e n t forms. In Chapter I I I a .Hamiltonian system i s considered i n the normalized form. (3) Z k = X k Z k + f k ( Z ) , k = 1,2,...,2n and Z « ( z i * Z 2 , . . . > Z 2 n). where f k are power series beginning with quadratic terms and two p a i r s of X's are purely imaginary. A p e r i o d i c sol u t i o n of ( 3 ) i s constructed i n the form (4) Z R = 2 ^ * k = l , 2 , . . . , 2 n . where cp^ i s a homogeneous polynomial of degree I i n terms of four time dependent variables. GRADUATE STUDIES Vector spaces and the Theory of Matrices Algebra I Theory of Functions of a Real Variable Theory of Functions I Measure and Integration Topology Nonlinear D i f f e r e n t i a l Equations II Theory of Functions II R. Westwick R. C. Thompson D. Derry D. Bures D. W. Bressler S. Cleveland E. Leimanis R. Cleveland ( i i ) ABSTRACT The subject matter of t h i s t h e s i s c o n s i s t s of a q u a l i -t a t i v e study of the s t a b i l i t y and asymptotic s t a b i l i t y of the zero s o l u t i o n of c e r t a i n types of nonlin e a r d i f f e r e n t i a l equations, f o r a r b i t r a r y i n i t i a l p e r t u r b a t i o n s , and the con-s t r u c t i o n of a p e r i o d i c s o l u t i o n f o r a Hamiltonian system w i t h n( _> 2) degrees of freedom. The m a t e r i a l I s d i v i d e d i n t o three chapters. The s t a b i l i t y of the system (1) x = x h 1 ( y ) + ay, y = f ( x ) + y h 2 ( x ) w i t h some r e s t r i c t i o n s on the f u n c t i o n s h-^y), h 2 ( x ) and f ( x ) , i s discussed i n the f i r s t .chapter. I t turns out tha t some of the r e s u l t s proved by I.H. MUFTI ( [ l ] , [ 2 ] , [3]), f o r the systems (2) x = x h 1 ( y ) + ay, y = xhg(x) + by and (3) x = x h x ( y ) + ay, y = bx + y h 2 ( x ) become p a r t i c u l a r cases of our r e s u l t s f o r system (1). Con-sequently an answer i n the a f f i r m a t i v e has been given t o a problem proposed by I.H. MUFTI [1]. In the same chapter a g e n e r a l i z a t i o n t o the problem of M. A. AIZERMAN [ l ] f o r the case n = 2 i s given i n the form ( 4 ) x = f x ( x ) + f 2 ( y ) , y = ax + f 5 ( y ) . ( i i i ) This system has "been discussed f i r s t by a q u a l i t a t i v e method and second by c o n s t r u c t i n g a LYAPUNOV f u n c t i o n . In chapter I I , s t a b i l i t y of a q u a s i l i n e a r equation (5) x" + a f 1 ( x , x ) x + f 2 ( x , x ) x + bf-^(x) = 0 i s d iscussed, hy us i n g LYAPUNOVs second method. I t has been proved that i f ( i ) f ^ ( x ) e C, f g e C , f 1 e C 2 r e l a t i v e to x 2 2 ( i i ) f L ( x , y ) > 1, g(x,y) > 1, ^ < x f ^ ( x ) < , (a,b > 0) f o r a l l values of x and y = x ( i i i ) ag(x,y) - b f ^ ( x ) > [i2 > 0 ([x > 0 ) , f o r a l l x,y ( i v ) y3 xg(x,y) < 0 and |x| ( G ( x , y ) - l ) < |t± ,/ | l ( x , y j | , y ^ 0 (where G,g and w are defined i n Theorem 2.1) 2 2 (v) w(x,y) - 0 0 as r = Jx + y -» » then the zero s o l u t i o n of (5) 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 p e r t u r b a t i o n s . In the same chapter c e r t a i n equations of t h i r d order have a l s o been discussed f o r "complete s t a b i l i t y " . These equations are s p e c i a l cases of (5) and are more general than those considered by SHIMANOV [ l ] and BARBASHIN [ l ] . AIZERMAN's [ l ] problem f o r the case n = 2 i s ge n e r a l i z e d t o two d i f f e r e n t forms, one of which i s (6) x = f x ( x ) + a l 2 y + a ^ z y = a 2 J x + f 2 ( y ) + a ^ z z = a^ 1x + a^ 2y + a ^ z ( i v ) which i s more general than the forms considered hy V.A. PLISS [4] and N.N. KRASOVSKII [ l ] . Under a non-singular l i n e a r t r a n s f o r m a t i o n equations(6) assume^ the form (7) x = ^ ( x ) + ax, y = (J)2(y) + bz, z = ^ ( x ) + ^ ( y ) . I t has been proved that i f i n a d d i t i o n to the usual existence and uniqueness requirements, the c o n d i t i o n s ( i ) a > 0, b > 0, ( i i ) (^(x), (j)^(x) - i 0 f o r x 2. o, ( i i i ) <|)2(y), ^ ( y ) | 0 f o r y ^ 0, ( i v ) r x p y |J <()3(x)dx| - co as Ix| - c o 3 |J M y ) d y | - • as |y| - • o ^ o are f u l f i l l e d , then the zero s o l u t i o n of (7) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . In the t h i r d chapter a Hamiltonian system w i t h n (> 2) degrees of freedom i s considered i n the normalized form (8) z k = X k z 1 + f k ( z ) , k = l , 2 , . . . ,2n, and z = ( z 1 , z 2 , . . . , z 2 n ) , where f k are power s e r i e s i n z k beginning w i t h quadratic terms. A p e r i o d i c s o l u t i o n f o r system (8) i s constructed i n the form 03 ( 9 ) z k = £ c p u , k = 1,2,...,2n 1=1 where cp k^ i s a homogeneous polynomial of degree I i n terms of four time dependent v a r i a b l e s a, B, y, 6. C. L. SIEGEL [ l ] c o n s t r u c t s a p e r i o d i c s o l u t i o n i n terms of two v a r i a b l e s ? and U under the assumption t h a t the corresponding l i n e a r system has a p a i r of p u r e l y imaginary eigenvalues. Here i t i s assumed that the l i n e a r system possesses two d i s t i n c t p a i r s of p u r e l y imaginary eigenvalues and t h i s n e c e s s i t a t e s the c o n s i d e r a t i o n of four time dependent v a r i a b l e s i n the c o n s t r u c t i o n of the p e r i o d i c s o l u t i o n . A C K N O W L E D G E M E N T S T h e a u t h o r w i s h e s t o t h a n k h i s a d v i s o r D r . E . L E I M A N I S , f o r s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s a n d for g u i d a n c e , a d v i c e a n d e x t r e m e e n c o u r a g e m e n t g i v e n t h r o u g h o u t t h e a u t h o r * g r a d u a t e s t u d i e s a n d d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . T h a n k s a r e a l s o d u e t o D r . J . F S c o t t - T h o m a s f o r r e a d i n g t h e t h e s i s a n d f o r m a k i n g v a l u a b l e c o m m e n t s . T h e f i n a n c i a l a s s i s t a n c e g i v e n b y t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a t h r o u g h i t s s u m m e r g r a n t , a n d b y 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 I n t h e f o r m , o f G r a d u a t e F e l l o w s h i p , i s a p p r e c i a t e d . ( v i ) TABLE OF CONTENTS INTRODUCTION SOME PRELIMINARIES from the Q u a l i t a t i v e theory of Nonlinear D i f f e r e n t i a l Equations. page 1 CHAPTER I Sect i o n 1.1 1. 2 1.3 1.4 CHAPTER I I 1.5 S e c t i o n 2,1 S t a b i l i t y i n the l a r g e of c e r t a i n types of Autonomous systems of two D i f f e r e n t i a l Equations The problem of AIZERMAN The S t a b i l i t y of the System x - f ± ( x ) + f 2 ( y ) , y = ax + f ? ( y ) The S t a b i l i t y i n the l a r g e of the system x = xh^(y) + ay, y = f ( x ) + y h 2 ( x ) . The S t a b i l i t y i n the l a r g e of the system x = f-j_(x) + ay y = f 2 ( x ) + by The S t a b i l i t y i n the l a r g e of the system x = ax + f - L ( y ) , y = f 2 ( x ) + cy . 15 15 17 23 26 31 2. 2 S t a b i l i t y i n the l a r g e of c e r t a i n types of Autonomous systems of three D i f f e r e n t i a l Equations 37 S t a b i l i t y i n the la r g e of the equation x + a f 1 ( x , x ) x + f 2 ( x , x ) x + b f j ( x ) = 0 3b* S t a b i l i t y of s p e c i a l cases 45 ( v i i ) 2.3 G e n e r a l i z a t i o n of AIZERMAK's problem i n the case n = 3 t o the form x = f x ( x ) + a 1 2 y + a 3 ; ?z y = a 2 ]_x +.f 2(y)+ a ^ z z = a ^ x + a^ 2y + a ^ z 55 2.4 S t a b i l i t y of the system x = f-j_(x) + a 1 2 y + a-^z y = f 2 ( x ) + a 2 2 y + a g^z z = a ^ x -f a 5 2 y + f ^ ( z ) 62 CHAPTER I I I On a p e r i o d i c s o l u t i o n of a Hamiltonian system w i t h n degrees of freedom. 67 S e c t i o n 3.1 I n t r o d u c t i o n 67 3-2 The c o n s t r u c t i o n of a p e r i o d i c s o l u t i o n f o r a Hamiltonian sys tems, i n the case of two p a i r s of p u r e l y . 70 imaginary r o o t s . 3.3 Convergence of the s o l u t i o n Bo BIBLIOGRAPHY 89 INTRODUCTION I n the f i r s t two chapters, of t h i s t h e s i s we d i s c u s s the the s t a b i l i t y of Autonomous systems of two and three d i f f e r e n t i a l equations. I n the t h i r d chapter i s constructed a p e r i o d i c s o l -u t i o n i n terms of f o u r v a r i a b l e s , of a Hamiltonian system w i t h n (n_>2/, degrees of freedom. I n recent years there has a r i s e n a considerable i n t e r e s t i n the theory of s t a b i l i t y of motion. The theory created i n the n i n e t i e s of the l a s t century by A. M. LYAPUNOV has found wide a p p l i c a t i o n i n various f i e l d s of p h y s i c s and technology. There are s e v e r a l methods of d i s c u s s i n g the s t a b i l i t y of a system of d i f f e r e n t i a l equations, but the D i r e c t method pf LYAPUNOV i s becoming an important t o o l i n the hands of Engineers. Q u a l i t a t i v e i n v e s t i g a t i o n of i n t e g r a l curves was s t a r t e d by POINCARE and continued by many-famous mathematicians i n c l u d i n g G. D. BIRKHOFF, N. G. CETAEV, and others. I n 1950 N. P. ERUGIN , ( [ 1 ] , [2]) formulated a general theorem of q u a l i t a t i v e nature f o r the s t a b i l i t y f o r a r b i t r a r y i n i t i a l p e r t u r b a t i o n s f o r a system of two equations. We w i l l , make a frequent use of t h i s theorem i n the f i r s t chapter. V. A. PLISS [3] g e n e r a l i z e d ERUGIN's theorem to a system of n d i f f e r e n t i a l equations. During the l a s t f i f t e e n years many Russian mathematicians, i n p a r t i c u l a r , N. N. KRASOVSKII, E. A. BARBASHIN, V. A. PLISS, K. P. PERSID/KII and S. N. SHIMANOV have published many papers on the s t a b i l i t y of motion. I n the Chapter 1 we d i s c u s s the s t a b i l i t y i n the -large of •the f o l l o w i n g system of two d i f f e r e n t i a l equations * = x h 1 ( y ) + ay ( - d ) y = f ( x ) + y h 2 ( x ) ' " ^ f i r s t by q u a l i t a t i v e method and second by c o n s t r u c t i n g a LYAPUNOV f u n c t i o n . Under various r e s t r i c t i o n s on the f u n c t i o n s h ^ ( y ) , f ( x ) and h 2 ( x ) , we w i l l prove that the zero s o l u t i o n of the above system i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . I t w i l l be shown that some of the r e s u l t s proved by I . H. MUFTI ( [ 1 ] , [ 2 ] , [ 3 ] ) > f o r the systems x = xh ]_(y) 4- ay a n d x = xh 1(y) + ay y = x h 2 ( x ) + by f = bx + y h 2 ( x ) 2. can be deduced e a s i l y from our r e s u l t s . Thus, t o some extent, we have solved the problem proposed by I. H. MUFTI [ l ] . In the same chapter, we w i l l a l s o d i s c u s s the s t a b i l i t y of the system x = f ]_(x) + f 2 ( y ) y = ax +• f ? ( y ) , which may be considered as a g e n e r a l i z a t i o n of the problem of AIZERMAN [1] f o r n = 2. In a d d i t i o n , a r e s u l t proved by MUFTI [2] w i l l be generalized. In Chapter I I the s t a b i l i t y of a q u a s i l i n e a r equation (5) x + af^ x - j - x j x + f 2 ( x , x ) x + bf-j(x) = 0 i s d iscussed, by usi n g LYAPUNOVs second method. I t has been proved t h a t i f i p ( i ) f-j(x) e C, f 2 e C , f-^ e C , r e l a t i v e t o x , ( i i ) f x ( x , y ) > 1, g(x,y) > 1, Is < x f ^ ( x ) _< |- , (a,b > 0) f o r a l l values of x and y = x, ( i i i ) ag(x,y) - b f ^ ( x ) > u 2 > 0 (n > 0), f o r a l l x,y , ( i v ) yS xg(x,y) < 0 and |x| (G(x,y)-1) < -|^  A / | l ( x , y ) | , y ^ 0 (where G,g and w are defined i n Theorem 2.1), 2 2 (v) w(x,y) as r = Jx + y - » , then the zero s o l u t i o n of (5) 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 p e r t u r b a t i o n s . In the same chapter, c e r t a i n equations of t h i r d order have a l s o been discussed f o r "complete s t a b i l i t y " . These equations are s p e c i a l cases of (5) and are more general than those considered by SRTMANOV [ l ] and 3. BARBASHIN [ l ] . The problem of AIZERMAN f o r a system of th r e e equations x = f (x) + a l 2 y + • a 1 5 z * y = a 2 1 x + a 2 2 y + a 2 j Z z = a-znx + a.-.y + a-,-,z 31 32 33 was d i s c u s s e d e x t e n s i v e l y by V.A. PLISS [ 4 ] . We w i l l d i s c u s s the g e n e r a l i z a t i o n of t h i s problem i n the f o l l o w i n g forms, X f - l ( x ) + a l 2 y + a l 3 2 ( i ) 0 y = + f 2 ( y ) + a 2 ^ z • z = a 3 l x + a 3 2 y + a 3 3 z and • X = f x ( x ) + a 1 2 y + a 1 3 z ( i i ) y ' = f 2 ( x ) + a 2 2 y + a2-^z • z 4- a 3 l x a 3 2 y + Under v a r i o u s assumptions we w i l l reduce the above systems t o some simpler forms. T h i s w i l l be achieved by means of non-s i n g u l a r l i n e a r t r a n s f o r m a t i o n s . Afterwards, s e v e r a l c o n c l u s i o n s w i l l be drawn by a p p l i c a t i o n of LYAPUNOVs D i r e c t Method and Q u a l i t a t i v e i n v e s t i g a t i o n s . A H a m i l t o n i a n system w i t h n (h >, 2) degrees of freedom w i l l be considered i n the t h i r d chapter. I t w i l l be assumed t h a t the system i s i n the nor m a l i z e d form z k = X k z k + f k ( z ) > (k=l,2,...,2n) where z = ( z ^ , z 2 , . . . , z 2 n ) and f k ( z ) are power s e r i e s i n z^., b e g i n n i n g w i t h q u a d r a t i c terms. We w i l l l o o k f o r a s o l u t i o n of 4 . t h i s system i n the form 03 2 k * 2. ' ( k - l> 2 * . . . , 2 n ) <L=1 where cp^^is a homogeneous polynomial of degree l i n four time dependent variables a, p, Y> and 6. C. L. SIEGEL [ 1 ] assumes that the l i n e a r system corresponding to the above system has a pa i r of purely imaginary eigenvalues, and consequently the peri o d i c solution constructed by him depends on two time dependent variables Q and '"^  I f the l i n e a r system possesses two d i s t i n c t p airs of purely imaginary eigenvalues, then t h i s construction cannot be carried through. Thus the reason f o r constructing the periodic solution i n terms of four variables becomes obvious. 5. SOME PRELIMINARIES from the q u a l i t a t i v e theory of nonl i n e a r d i f f e r e n t i a l equations Let t denote the time and suppose that some p h y s i c a l system i s described by v a r i a b l e s y., where y^ depend on time and suppose y^ s a t i s f y d y i (0.1) -gr- = Y 1 ( y 1 , y 2 , . . . , y n , t ) , i«l,2,...,n. Assume y^ =x f ^ ( t ) i s some p a r t i c u l a r s o l u t i o n of ( 0 . 1 ) . I n order to study the behaviour of the s o l u t i o n s of the system (0.1) i n the neighborhood of the s o l u t i o n f . ( t ) , we perform the f o l l o w i n g change of v a r i a b l e s : (0.2) x± = y±- f ± ( t ) ,1=1,2,...,]!. We w i l l c a l l x. = 0 or y =* f. (t) the unperturbed motion. Since the v a r i a b l e s y.. s a t i s f y ( 0 . 1 ) , the new v a r i a b l e s x^ s a t i s f y the f o l l o w i n g d i f f e r e n t i a l equations: (0.3) x ± - . X i ( x 1 , x 2 , .. . , x n , t ) i - 1 , 2 , ...,n , (* = where X± - Y i ( x 1 + f 1 , . . . , x n + f n , t ) - Y i ( f ] _ , f 2,... , f n , t ) . F o l l o w i n g LYAPUNOV we w i l l c a l l the equations (0.3) the equations of the perturbed motion. I n the subsequent work the v a r i a b l e s y^ are suppressed and the v a r i a b l e s x. are t r e a t e d d i r e c t l y . The curve { f 1 ( t ) } i s always denoted by x. = 0 and i s c a l l e d the zero s o l u t i o n or the unperturbed s o l u t i o n . Thus without l o s s of g e n e r a l i t y we can assume th a t (0.4) x^(0,0,...,0,t) = 0, 1=1,2,...,n. The f u n c t i o n s w i l l be assumed t o be continuous w i t h respect to a l l t h e i r arguments and to s a t i s f y the c o n d i t i o n s of uniqueness i n the reg i o n 6. (0.6) - » < x| < » , t > 0 . For the sake of s i m p l i c i t y l e t us denote the vector ( x - ^ X g , ... ,x n) hy the s i n g l e l e t t e r x. A s o l u t i o n passing through a p o i n t ( x ^ Q j X g Q , ' x n n ) o f t h e ( xi>x2>—*xn^ s P a c e > a t time t =» t Q i s denoted hy x i ( x 1 Q , . . . , x n Q , t Q i i t ) , or simply hy x ( x Q , t Q , t ) . Consequently, we can w r i t e the system (0.3) i n the form (0.7) x » X ( x , t ) where X ( 0 , t ) « 0 . and (0.7) admits the t r i v i a l s o l u t i o n x ( t ) s, 0. D e f i n i t i o n 0.1: The zero s o l u t i o n x ( t ) » 0 of the system (0.7) i s c a l l e d s t a b l e (at t • t ) i n the sense of E5TAPXM0V, i f f o r a r b i t r a r y p o s i t i v e e > 0 there e x i s t s a 6 m 6 ( e , t Q ) such t h a t , whenever ||x0|| < 6 , where Hxoll 3 8 ( x i o + ••• + x n o ) 1 / 2 > t h e i n e ( i u a l i t y H v V ^ H < € i s s a t i s f i e d f o r a l l t > t > 0. — o D e f i n i t i o n 0.2: The zero s o l u t i o n x ( t ) - 0 of the system (0.7) i s c a l l e d a s y m p t o t i c a l l y s t a b l e and the re g i o n G 6 = i x : ||x|| < 6} of the x-space i s s a i d to l i e i n the regi o n of a t t r a c t i o n of the po i n t x =» 0 (at t » t ), provided that the c o n d i t i o n s of d e f i n i t i o n 0.1 are s a t i s -f i e d , and provided f u r t h e r that ( I ) 11m x(x , t ,t) - 0 ( i i ) x ( x Q , t ,t) e f , t > t Q t-*» f o r a l l values of x Q l y i n g i n Gfi. Here ^~ i s some sub-regi o n of G where are defined and continuous. D e f i n i t i o n 0.3: I f , however, l i m x(x , t ,t) => 0 t-°° ° ° independent of the choice of xQ or tQ J> 0, then the unperturbed motion i s said t o be a s y m p t o t i c a l l y s t a b l e i n the l a r g e . There are other types of s t a b i l i t i e s as w e l l ; e.g. uniform s t a b i l i t y , uniform asymptotic s t a b i l i t y , 7. structural s t a b i l i t y , stability of the mth order In'" the sense of G. D. BSRKHOFF, orbital stability, etc. And there are different definitions of "s t a b i l i t y " which have l i t t l e to do with one another, though there are intricate connections between them. For f u l l analysis we can refer to WXNT.NER [1] , J . L . MAS SERA [1] , I. G. MALKIN [2] and N . N.' KRASOVSKII [ 4 ] . In the present work we w i l l only discuss s t a b i l i t y "in the large". A. M. LYAPUKTOV divided the methods of solving the problem of stability into two categories. In the f i r s t category he placed those methods which are reduced to the direct considera-tion of disturbed motion, i.e., to the determination of the gen-eral or particular solution of the corresponding differential equations. The aggregate of these methods i n the f i r s t category was termed by LYAFTTNW as the f i r s t method. In some cases the problem of stability can be solved without actually finding the particular and general soltuions of equations of the perturbed motion, but by finding certain functions of t,x 1,...,x n, which possess special properties. As an example we may consider the well-known theory of Lagrange on stability of equilibrium when the force function i s a maximum. This stability i s assured by the existence of a force function possessing special properties. The . v aggregate^ of these methods of the second category LYARUlfOV named the second method. For a long time this method was considered to have only theoretical value and was almost forgot-ten. The reversal of this situation i s now so complete that It has become one of the most effective tools i n the hands of Engineers and Physicists. Let V(x,t) denote a continuous scalar function of x,t which has continuous partial derivatives of the f i r s t order i n some region of x-space that contains the unperturbed solution x m 0 for a l l t > 0, and V(0,t) - 0. 8 . D e f i n i t i o n 0.4: I f the i n e q u a l i t y V (x,t) > 0 [ or V (x,t) _< 0] holds f o r a l l x i n p and f o r a l l t > 0, the f u n c t i o n V ( x , t ) i s said to he s e m i - d e f i n i t e i n P . D e f i n i t i o n 0.5: A f u n c t i o n W(x), which does not depend e x p l i c i t l y on t , i s s a i d t o he d e f i n i t e i n p i f i t i s p o s i t i v e - d e f i n i t e or n e g a t i v e - d e f i n i t e i n p , i . e . , i f f o r a l l x e p , x ft 09 the r e l a t i o n W(x) > 0 [or ¥(x) < 0] hold s. D e f i n i t i o n 0.6: The f u n c t i o n V ( x , t ) i s c a l l e d p o s i t i v e -d e f i n i t e [ n e g a t i v e - d e f i n i t e ] i f V( x , t ) j> W(x) f o r x. e f , t > 0 [V(x,t) _< -W(x), f o r x e T, t > 0] holds f o r some p o s i t i v e - d e f i n i t e f u n c t i o n W(x). D e f i n i t i o n 0 . 7 : A f u n c t i o n V ( x , t ) admits an i n f i n i t e l y s m a l l upper bound i n p , provided there e x i s t s a continuous f u n c t i o n W(x) 3 W(0) - 0 and the r e l a t i o n s | V ( x , t ) | < W(x) hold f o r x € p , t > 0. D e f i n i t i o n 0 . 8 : I f f o r every M > 0 there e x i s t s a number N >C ? f o r || x|| >_ N, t > 0 |V(x,t)| > M, then V(x,t) i s sai d t o be i n f i n i t e l y .large. D e f i n i t i o n 0 .9: A d e f i n i t e f u n c t i o n v ( x , t ) , the E u l e r i a n d e r i v a t i v e of which w i t h respect to the perturbed equations i s ©44feex s e m i - d e f i n i t e of s i g n opposite to t h a t of v(x,t) ciy*. n « j rlnn-M-on 1 1 y- r>rpua 1 | r n - g g L T Q ^ J T S C a l l e d a LYAPUNOV f u n c t i o n or a v - f u n c t i o n . n /7_, .. . .. ... , dV \ 3 7 &V \ ( E u l e r i a n aenva'cive oi y i s - 7 - = / — X. + ~s~ ). d t L 3x. i at ' 1=1 x We s t a t e here some c l a s s i c a l theorems on s t a b i l i t y and asymptotic s t a b i l i t y f o r the system^-( 0 . 8 ) x = X ( x ) , X = ( X 1 , . . . , X N ) - , where are continuously d i f f e r e n t i a b l e f u n c t i o n s of the v a r i a b l e s x ^ j . . . , x i n the re g i o n -« < x^ < » and X(0) = 0. Theorem 0 . 1 ; I f there e x i s t s i n some neighbourhood G of o r i g i n 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 7(x) such that V(x) = (grad V ) • X i s n o n - p o s i t i v e , then the zero s o l u t i o n of ( 0 . 8 ) i s s t a b l e . (grad V e x i s t s since V ( x ) has f i r s t p a r t i a l d e r i v a t i v e s ) . Theorem 0 . 2 : I f i n Theorem 0.1 V ( x ) i s n e g a t i v e - d e f i n i t e , then the zero 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 . STABILITY ON THE BASIS OF ABRIDGED EQUATIONS Consider the nonlin e a r autonomous system n ( 0 . 9 ) x 1 = ^ P i j X j + X ± ( x ) , i = l , 2 , . . . , n , J-1 where p. . are constants and X. are power s e r i e s w i t h at l e a s t i j . 1 second degree terms. The f o l l o w i n g theorem can be proved (MINORSKY [ l ] ) on the b a s i s of the v a r i a t i o n a l (abridged) equations (0 .10) x ± = 2. P i ;j xj-> i = l , 2 , . . . , n J=l and the equation 10. (0.11) D(X) P n - X p 1 2 '21 "^22"" ^ * * * 'in >2n 0 'nl 'n2 Pnn" 1 Theorem 0 . 3 : I f a l l the roo t s of (0.11) have negative r e a l p a r t s then the zero s o l u t i o n of ( 0 . 9 ) i s asymp-t o t i c a l l y s t a b l e whatever the terms X^ may be. We know t h a t the converse of the theorem 0.2 i s t r u e . I n other words, f o r asymptotic s t a b i l i t y i t i s necessary and s u f f i c i e n t t h a t there must e x i s t 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) having 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 respect t o time along a l l the motions of the system ( 0 . 8 ) . E. A. BARBASHTN and N. N. KRASOVSKII [1] showed by c o n s t r u c t i n g an example t h a t the existence of the Lyapunov f u n c t i o n does not ensure the asymptotic s t a b i l i t y i n the l a r g e . I n fact., they considered the system 2x ( 1+x 2) 2 2y f „ 2x (1+x2) 2 (1+x 2) 2 2 x 2 and used V(x,y) « y, + as the v - f u n c t i o n which g i v e s 1+x^ 4 x 2 4 y 2 V ( l + x 2 ^ " ( 1+x 2) 2 I n order t o e x h i b i t a domain of i n s t a b i l i t y i n the xy-plane a curve (Y) given by y * 2 + — i p — was considered. By computing 1+x -x and y along t h i s curve i t Was found t h a t 2x ,. 2 (1+x ) 1+x' ( 1+x 2) 2 "% ( 4 + " ( 1+x 2) 2 13. 2^r = - 2y 2cp(x,y), and we can see t h a t Jjr < 0 f o r y ^ 0 a n d H - O.for y - 0 I f we take f o r M the x - a x i s , i . e . , y - 0, then c l e a r l y an a r b i t r a r y i n t e r s e c t i o n M n(V - C, C > 0) does not c o n t a i n any p o s i t i v e h a l f - t r a j e c t o r y of the system (0.13). I f we f u r t h e r assume t h a t ( 0 , 1 5 ) j f ( x ) dx—*«> f o r | x | : - 7 > < » , then the zero s o l u t i o n of the system (0.13) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . I n 1950, N. P. ERUGIN [1] proved the f o l l o w i n g theorem f o r the system ( ° ' 1 6 ) | f - P ( » , y ) . - . | | - « ( x . y ) Theorem 0.7: (ERUGIN.) we assume th a t ( i ) the p o i n t (0,0) i s the only p o i n t of e q u i l i b r i u m f o r the system (0.16). ( i i ) the unperturbed motion x « 0 • y i s a s y m p t o t i c a l l y s t a b l e and consequently any motion s t a r t i n g i n a re g i o n (0.17) G e: x 2 + y 2 < e, possesses the property x(t)—-»0, y(t)—>0 as t — ( i i i ) A s t r a i g h t l i n e L(0,») going t o i n f i n i t y from the p o i n t (0,0) i s i n t e r s e c t e d by the motions i n one d i r e c t i o n only f o r t«-*•».. ( i v ) The motions having bounded p o l a r angles are bounded, (v) There are no p e r i o d i c motions, Then a l l the motions possess the property x(t)—>0, y ( t ) - ^ o . The above theorem has been g e n e r a l i z e d t o the case of a system of n equations by V. A. PLISS [ 4 ] , D e f i n i t i o n 0. 10: A p o i n t q on the x-space i s c a l l e d an w - l i m i t p o i n t of a t r a j e c t o r y f ( p , t ) , p a s s i n g through a p o i n t P, of the system (0 . 8 ) i f 3 a sequence t 1 , t 2 , . . . , t -j?+» 1 ' sv.cn that H I T: p ; f (p. t V / } , q} - 0 , where o (f (p, ) q ) stands f o r the distance between the po i n t f ( p , t ) of the t r a j e c t o r y f ( p , t ) and the point q. A point q i s c a l l e d an a - l i m i t p o i n t of a f ( p , t ) i f there e x i s t s a sequence . t ^ , t . 0 . t , . . . w i t h t„ - « such that l i m p (f (p, t r , ) , gj = 0. . Ti -> w D e f i n i t i o n 0 . 1 1 ; An i n v a r i a n t set i s a set w i t h the property that a s o l u t i o n s t a r t i n g i n the set remains i n s i d e i t f o r a l l t p o s i t i v e or negative. The f o l l o w i n g theorem i s due to L a S a l l e [3'J. Theorem ChS; Let V ( x ) be a s c a l a r f u n c t i o n w i t h continuous f i r s t p a r t i a l d e r i v a t i v e s f o r a l i x. Suppose that V(x)>0 f o r a l l x £ 0 and V(x) _< 0 . Let S he the set defined by V(x) = 0 and l e t M denote the l a r g e s t i n v a r i a n t set i n E. Then every s o l u t i o n boimded ( i n the f u t u r e ) approaches M as t -» » . ROUTH-HURWITZ CRITERION; We consider the f o l l o w i n g polynomial f(z') = zn -r a-, z'' ' -f — + av • ' J_ n w i t h r e a l c o e f f i c i e n t s . Put D, = a, .and a 5 a 2 k - l a;., • a 2 k - 2 a 2 k - 3 k—c }3 i»•« ,n 0 0 0 a,, w i t h a.. = 0 f o r j > n. I f a l l determinants Dfc. are p o s i t i v e , k = 1,2,...,n, then a l l the.zeros of f ( z ) have negative r e a l p a r t s . 1 5 . CHAPTER I STABILITY IN THE LARGE OF CERTAIN TYPES OF AUTONOMOUS SYSTEMS OF,TWO DIFFERENTIAL EQUATIONS 1 . 1 The problem of AIZERMAN. I n 'his study of servo-mechanisms M. A. AYZERMAN [ 1 ] formulated the f o l l o w i n g problem. Let there be given a system of l i n e a r d i f f e r e n t i a l equations dx^ n cMr n t 1 ' 1 ) o t ~ = j l l a l j x j + a?k,= J = l a i O X J ' : i = 2^>--->n-Suppose th a t f o r given constants a. . ( i , j = l , 2 , . . . , n ) and f o r an a r b i t r a r y value of the constant a from the i n t e r v a l a < a < B,.„all the root s of the c h a r a c t e r i s t i c equation of the system ( 1 . 1 ) have negative r e a l p a r t s . I n the svstem ( 1 . 1 ) l e t a x k be replaced by an a r b i t r a r y c o n t i n -uous f u n c t i o n f ( x k ) which s a t i s f i e s the f o l l o w i n g c o n d i t i o n s : ( 1 . 2 ) f ( x k ) = 0 f o r x k = 0, a x 2 < x f c f ( x k ) < p x 2 , x k ± 0. I t i s required to f i n d out whether or not the t r i v i a l s o l u t i o n x ( t ) = 0 i s now a s y m p t o t i c a l l y s t a b l e i n the l a r g e . I t was pointed out by V. A. PLISS [2] that the answer to t h i s prQblem i s i n the negative. According t o I . H. MUFTI [ 1 ] , the i n t e r e s t i n the problem i s revived i f we ask ourselves the f o l l o w i n g questions: For what values of a^j i s the answer to t h i s problem In the a f f i r m a t i v e , and f o r what values i s the answer i n the negative? I f the s o l u t i o n i s not a s y m p t o t i c a l l y s t a b l e i n the l a r g e under ' the g e n e r a l i z e d HURWITZ c o n d i t i o n s f ( 0 ) = 0, a x 2 < x k f ( x k ) < P x 2 , x k f 0 what a d d i t i o n a l assumptions should be made on f ( x k ) so that the zero s o l u t i o n becomes a s y m p t o t i c a l l y s t a b l e i n the large? This problem has been considered by many authors i n the 16. case n - 2 i n c l u d i n g N. N. KRASOVSKII, I . H. MUFTI arid'V. A. PLISS. For n > 3 we know only t h a t i f the c o e f f i c i e n t s a^. are connected hy c e r t a i n r e l a t i o n s the problem has a p o s i t i v e s o l u t i o n ("V. A. PLISS [1], S. N. SHIMANOV [1]). We w i l l here consider a ge n e r a l -i z a t i o n of t h i s problem i n the case n - 2 , i n the f o l l o w i n g form: (1.3) * - f x ( x ) + f 2 ( y ) y - ax + f 5 ( y ) where f ^ , f 2 and f ^ are continuous f u n c t i o n s f^(0) - f 2 ( 0 ) -f^(0) - 0 and 'a' i s a constant. L e t t i n g : . h l ( x ) „ _ J ^ _ _ , x f 0, h 2 ( y ) - - ~ r - , h 3 ( y ) - -2—, y 0 and assuming th a t (1.4) a < 0, a h x ( x ) > 0 f o r a l l x f 0, h 2 ( y ) > 0, h^(y) _< 0 f o r a l l y f 0, we prove the f o l l o w i n g theorem Theorem 1.1. I f i n a d d i t i o n t o the assumptions(1.4) we assume that y y h 2 ( y ) d y -» « as |y| - then the zero s o l u t i o n of the system (1.3) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . Proof. We consider the f o l l o w i n g V - f u n c t i o n 2V - 2J^yh2(y)dy - a x 2 I n view of the assumptions (1.4) i t i s easy t o see t h a t 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 and I s i n f i n i t e l y l a r g e , i . e . , /~2 2~ "V - oo as r - ,/x +y -* * . We compute the d e r i v a t i v e of V w i t h respect t o time along a t r a j e c t o r y of the system (1.3)• 2V - 2 y h 2(y)§ - 2 a x § - 2 y h 2(y)[ax + y h 5 ( y ) ] - 2 a x[xh 1(x)+yh 2(y)] - 2 y 2 h 2 ( y ) h 3 ( y ) - 2 a x 2h 1(x) 17. O b v i o u s l y V < 0 f o r x ^ 0 , y ^ 0 and V C O f o r x = 0 . S i n c e i s f i n i t e a t e v e r y p o i n t o t h e r t h a n t h e o r i g i n on the y - a x i s , t h e r e f o r e t h e y - a x i s cannot c o n t a i n any e n t i r e t r a j e c t o r y of system ( 1 . 3 ) . C o n s e q u e n t l y , we Can use Theorem 0 . 6 t o complete our p r o o f . The purpose o f t h e a s s u m p t i o n t h a t G(y) = J y h g ( y ) d y -» 0 0 as |yj -»"*, i s t o ensure t h e boundedness i n t h e f u t u r e o f a l l s o l u t i o n s of system ( 1 . 3 ) . I f , however, G(y) / 1 4 as |y| -• °* t h e n we can no l o n g e r c l a i m V t o be i n f i n i t e l y l a r g e and t h e r e -f o r e Theorem 0 . 4 ( o r 0 . 6 ) cannot be a p p l i e d . T h i s s i t u a t i o n i s d i s c u s s e d i n t h e n e x t s e c t i o n . 1 . 2 The s t a b i l i t y i n t h e l a r g e of t h e system x = f 1 ( x ) ' + f 2 ( y ) * y = ax + f ^ ( y ) u s i n g q u a l i t i a t i v e methods. We w i l l d i s c u s s t h e system ( 1 . 3 ) by a q u a l i t a t i v e method and show t h a t . t h e r e q u i r e m e n t t h a t r y y h p ( y ) d y -»' •» as |y| - «• can be r e l a x e d . J o * 18. L e t us assume (1.5) a < 0 , h 2 ( y ) > 0 , h-^x) _< c < 0 , c f i n i t e and b.^(y) < 0 . Consider the curves x h 1 ( x ) + yhg(y) » 0 and ax + yh^(y) - 0 . The o r d i n a t e s of these curves are r e s p e c t i v e l y given hy x N ( x ) > s y l * "h (y ) according as x ?< 0 y 2 " ~ h X ( y ) > < : 0 a c c o r < 3 i n g a s x ^ 9 We assume, without l o s s of g e n e r a l i t y t h a t the curves under c o n s i d e r a t i o n are s i n g l e - v a l u e d . The shapes of the curves are shown i n P i g . 1. I t i s easy to see that x - x h 1 ( x ) + y h 2 ( y ) < 0 t o the r i g h t of the curve xh-^x) + y h 2 ( y ) - 0 x = " " > 0 " " l e f t " " " 11 y - ax + yh^(y) < 0 " " r i g h t " " " ax + y h j ( y ) » 0 y - " " > 0 " " l e f t " " " " I f we change t o p o l a r coordinates, i . e . i f we l e t x » r coscp , y «• r sincp , then the signs of f and i n a l l the e i g h t regions i n t o which the plane i s d i v i d e d hy the two curves and the axes, are as f o l l o w s . I n the regions (1,5) r < 0 and $ may he 2. 0 " " " (2,6) f may be 1 0 and cp < 0 (3.7) t < 0 and cj> may be >. 0 (4.8) f may be 2. 0 and cp < 0 The 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 the sense of LYAPUNOV as i s c l e a r from the ex i s t e n c e of V - f u n c t i o n , from which i t i s a l s o evident t h a t there does not e x i s t any p e r i o d i c s o l u t i o n . The absence of any p e r i o d i c s o l u t i o n may a l s o be seen by the c r i t e r i o n of BENDIXSCN. Our aim i s t o use 19. ERUGIN's Theorem. For the l i m e t i ( 0 , » ) appearing i n t h a t theorem we may take the p o s i t i v e semi-axis, as a l l the motions cut i t i n the same d i r e c t i o n . ^ We need only consider the regions (2), (4), (6) and (8), where we w i l l show th a t there are no motions w i t h bounded p o l a r angles, i . e . , any motion s t a r t i n g i n o r e n t e r i n g any of these regions must leave t h a t r e g i o n a f t e r a s u f f i c i e n t time. Now rip - - s i n cp * [ x h 1 ( x ) + y h 2 ( y ) ] + cosccp- [ax + y h 5 ( y ) ] I n the r e g i o n (2), x may become i n f i n i t e (y remains bounded). Consequently, i n t h i s r e g i o n cp _< k < 0 f o r a c e r t a i n constant k^and cp decreases monotonically w i t h the Increase of time. Hence any motion s t a r t i n g i n region (2) or e n t e r i n g I t must leave t h i s r e g i o n and enter r e g i o n (1). The same argument can be a p p l i e d t o re g i o n (6). Now by assumption.C i s the l e a s t upper bound of h^(x) and C i s f i n i t e . L e t h ^ x ) « C + ty(x), where i|r(x) <_ 0. We consider the expression ax - cy and examine i t s r a t e of change along a t r a j e c t o r y of our system (1.3). (1-6) |^(ax-cy) - a ( x h x ( x ) + y h 2 ( y ) - c(ax + yh^(y)) » a ( x ( c + ^(x)) + y h 2 ( y ) - c(ax + yh^(y)) • axijr(x) + ayh 2(y) - cyh^(y) , 0 i n r e g i o n (4). Now l e t us consider a s t r a i g h t l i n e ax-cy « A, A > 0. T h i s s t r a i g h t l i n e cuts the curve ax + yh^(y) » 0 f o r every value of A, i n the second quadrant. Consider the r e g i o n enclosed by the negative semi-x-axis, the l i n e ax-cy » A and the.curve ax+yh^(y)«0. Any motion s t a r t i n g i n or e n t e r i n g t h i s r e g i o n cannot cut the l i n e ax-cy •> A, because of (1.6). Consequently i t must cross the curve ax + yh^(y) » 0 and enter r e g i o n (3)• The same r e a -soning can be c a r r i e d out f o r r e g i o n (8). I n regi o n (1), (3), (5) and (7) i t I s c l e a r t h a t any motion w i t h bounded p o l a r angle i s bounded. Since a l l the c o n d i t i o n s of theorem 0..7 are f u l f i l l e d t h e r e f o r e we have proved the f o l l o w i n g theorem. 21. Theorem 1.2. Under the assumptions (1.5)> the zero s o l u t i o n of the system ( 1 . 3 ) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . 1 . 3 . The s t a b i l i t y i n the l a r g e of the system (1.7) x m xh x(y) + ay y = f ( x ) + y h 2 ( x ) I . H. MUFTI [1] considered the systems (1.8) ± = xh x(y) + ay, y = x h 2 ( x ) + by and (1.9) x = x h 1 ( y ) + ay, y = bx + y h 2 ( x ) and proposed the f o l l o w i n g problem: I s i t p o s s i b l e to construct a LYAPUNOV f u n c t i o n f o r the system (1.7), of which the systems (1.8) and (1.9) are p a r t i c u l a r cases? We w i l l answer h i s ques-t i o n i n the a f f i r m a t i v e and show by c o n s t r u c t i n g a LYAPUNOV f u n c t i o n and a l s o by a q u a l i t a t i v e method that some of the r e s u l t s f o r the systems (1.8) and (1.9) can be deduced from our r e s u l t s f o r the system (1.7). Let h,(x) = --(xy, f o r x |= 0, and assume that j x (1.10) / ah^(x) < 0, h 1 ( y ) < 0 and h 2 ( x ) _< 0 f o r a l l x =}= 0 I and y =|= 0 and \ x j j.xh^(x)dx -» oo f o r |x| -• », a < 0 . We consider the f o l l o w i n g f u n c t i o n : X- 2 2V = 2j^xh^(x)dx - ay Obviously 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 and i s i n f i n i t e l y l a r g e . Computing the t o t ^ a l d e r i v a t i v e of V w i t h respect t o time along a t r a j e c t o r y of the system (1.7) we see t h a t 2V = 2 x h 3 ( x ) g - 2ay^| = 2xh^(x)[xh 1(y) + ay] - 2ay[xh^(x) + y h 2 ( x ) ] «- 2 x 2 h 1 ( y ) h 3 ( x ) - 2 a y 2 h 2 ( x ) 22. I t can be e a s i l y checked t h a t V < 0 f o r x , f 0, y <J» 0 a n d v" • 0 at the o r i g i n and p o s s i b l y on the l i n e x - 0, but the l i n e x - 0 does not contain any p o s i t i v e h a l f - t r a j e c t o r y of the system (1.7)• Consequently, by a p p l y i n g theorem 0.6 we have proved the f o l l o w i n g theorem. Theorem 1.3. Under the assumptions (1.10) the zero s o l u t i o n  of the system (1.7) i s a s y m s t o t i c a l l y s t a b l e i n the  l a r g e . Example. Consider the f o l l o w i n g system: We may take h^(y) » T e y < 0, h 2 ( x ) - -x < 0 and h^(x) - x and by a p p l y i n g theorem 1.3 we can d i s c u s s the s t a b i l i t y of t h i s system. 1. For the system (1.8) I . H. MUFTI [1] proved t h a t i f ah 2(x) < 0 f o r x 0, b < 0 and h-^y) < 0 f o r y + 0, then the zero s o l u t i o n of t h i s system 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 p e r t u r b a t i o n s . As we w i l l see i n our q u a l i t a t i v e d i s c u s s i o n , the c o n d i t i o n ••! can be. r e l a x e d , and then MUFTI' s,;.assumptions and h i s ..'result f o l l o w from ours. 2. He f u r t h e r assumes th a t i f b + h ^ y ) < 0, b h 1 ( y ) - a h 2 ( x ) > 0 f o r x f 0, y f 0, and i f e i t h e r b > 0 or h ^ y ) > 0, y f 0, then the zero s o l u t i o n of the system (1.8) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e provided 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 the s m a l l . x - -xe y+ ay, where a < 0. "5 2 y - x^ - yx Remarks. 23. His assumption b-fh-^y) <0 corresponds t o our h^(x)+h^(y) <0 " " bh 1(y)-ah 2(x)>0 " " " h 2 ( x ) h 1 ( y ) - a h ^ f x ) > 0 I f we suppose that b<0, h^'y^O and omit the requirement of the asymptotic s t a b i l i t y i n the s m a l l , then h i s r e s u l t f o l l o w s from our r e s u l t . 3. I n theorem 2.4 [2] he assumes b=0, h-^y^O, -ah2(x)>0. C l e a r l y h i s theorem i s a p a r t i c u l a r case of our theorem. 4. For the system (1.9) he proved the theorem 3.1 [2]. He assumes ab<0, h-^y^O when y=}=0, h2(x)<0 f o r x|<), w i t h s t r i c t i n e q u a l i t y i n a t l e a s t one of the l a s t two i n e q u a l i t i e s . I t i s easy to check that h i s r e s u l t i s a p a r t i c u l a r case of our r e s u l t . The S t a b i l i t y of the system (1.7) by using a q u a l i t a t i v e method. We now d i s c u s s our system (1.7) by a q u a l i t a t i v e method and e s t a b l i s h our c l a i m made i n remarks 1 and 2. Let us assume that (1.11) a < 0, h^(x) > 0, x =|= 0, h 2 ( x ) < 0 f o r x |= 0 and h x ( y ) < 0, y ± 0. and consider the curves x h 1 ( y ) + ay = 0 and xh^(x) + y h 2 ( x ) = 0. The ordinates bf these curves are r e s p e c t i v e l y given, by x hiW > < y. = < o according as x > 0 xh,(x) >  y2 ~ " h (x) < 0 according as x < 0 The shape of each of these curves i s shown i n F i g . 2. The curves and the axes d i v i d e the xy-plane i n t o e i g h t .regions. ic = xh 1(y) + ay <t 0 t o the r i g h t of the curve xh 1(y) + ay = C x = " > 0 " " l e f t " " " " " y = x h 3 ( x ) + h 2 ( x ) y > 0" " r i g h t " xh^(x) + yhg(x) = 0 y = " " " l e f t " " " " Using the p o l a r coordinates r and 0 we see that the signs of r and i n a l l the ei g h t regions are as f o l l o w s : FIG. 2-2 5 . I n the regions ( 1 , 5 ) t may be 2. 0 , cp > 0 " " 11 ( 2 , 6 ) t < 0 andS may be > 0 " " " ( 3 , 7 ) t may be ^  0 , $ > cf " " " ( 4 , 8 ) f < 0 and $ may be ^  0 < The zero 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 the s m a l l , as i s c l e a r from the existence of V - f u n c t i o n constructed e a r l i e r . We w i l l show t h a t the motions w i t h bounded p o l a r angles are bounded. I n regions ( 2 ) , ( 4 ) , ( 6 ) and' ( 8 ) f < 0 t h e r e f o r e any motion w i t h bounded p o l a r angle i s n e c e s s a r i l y bounded. Consequently we consider only regions ( 1 ) , ( 3 ) , ( 5 ) and ( 7 ) . Here i t w i l l be shown th a t there are no motions w i t h bounded p o l a r angles i . e . , any motion s t a r t i n g i n or e n t e r i n g these regions must leave these regions w i t h i n c r e a s e of time. We w r i t e cp m i [ - s i n c p ( x h 1(y) + ay) + coscp (xh^(x) + y h 2 ( x ) ) ] Le t us suppose th a t a motion s t a r t i n g i n or e n t e r i n g the re g i o n ( 1 ) does not leave t h i s r e g i o n as t - ». Since x decreases and y in c r e a s e s w i t h i n c r e a s e of t (y may become i n f i n i t e l y l a r g e ) , Consequently cp ^  k > 0 f o r some constant k, which i m p l i e s t h a t cp in c r e a s e s monotonically w i t h time and cannot remain bounded. Consequently any motion s t a r t i n g i n or e n t e r i n g r e g i o n ( 1 ) leaves t h i s r e g i o n and enters r e g i o n ( 2 ) a f t e r a s u f f i c i e n t time. The same argument can be c a r r i e d , out f o r r e g i o n ( 5 ) . I f we assumethat hg(x) _< c < 0 , w i t h c f i n i t e , we have h g ( x ) - c + t ( x ) , where \[f(x) _< 0 . Now consider the expression cx - ay and compute i t s r a t e of change along an a r b i t r a r y t r a j e c t o r y of the system ( 1 . 7 ) . ( 1 . 1 2 ) ^ - ( c x - ay) - c ( x h 1 ( y ) + ay) - a(xh^(x) + (c + t ( x ) ) y ) « e x h ^ y ) + acy - axh^(x) - acy - ayty(x) - x ( c h x ( y ) - ah^(x)) - ayi{r(x) < 0 i n the re g i o n ( 3 ) and > 0 i n the regi o n ( 7 ) . 26. We consider a s t r a i g h t l i n e cx - ay • X. T h i s l i n e cuts the curve xh,(y) + ay • 0 f o r a l l values of I. L e t i > 0 and consider the region between the p o s i t i v e semi-axis, the curve xh^(y) + ay - 0 and the l i n e cx - ay - I. A t r a j e c t o r y s t a r t i n g i n or e n t e r i n g r e g i o n (3) must cross xh^(y) + ay « 0. Since i t does not cross the l i n e cx - ay « X because of (1.12) and does not approach the o r i g i n i n view of cp > 0. Consequently we have ( i ) the zero 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 the sense of LYAPUNOV, ( i i ) t r a j e c t o r i e s w i t h bounded p o l a r angles are bounded, ( i i i ) there are no p e r i o d i c s o l u t i o n s , since the very e x i s t e n c e of a V«-function r u l e s out such a p o s s i b i l i t y ( t h i s i s a l s o c l e a r from BENDIXSON c r i t e r i o n , as f^Cxh^y) + a y ) + |y( x h ^ ( x ) + y h 2 ( x ) ) - h ^ y ) + h g ( x ) < 0,) and ( I v ) we can take f o r L(o,») the p o s i t i v e x - a x i s as a l l motions cut i t i n the same d i r e c t i o n f o r a l l t > 0. Consequently by ERUGIN's theorem we have proved the f o l l o w i n g Theorem 1.4. Under the assumptions (1.11) and the c o n d i t i o n that hg(x) • <j c < 0, c f i n i t e , the zero s o l u t i o n of the  system of the system (1.7) 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 . 1.4. The S t a b i l i t y i n the l a r g e of | | « f-^x) + a y / | j | . - f 2 ( x ) + b y The system (1.13) * . f x ( x ) + ay, y - f 2 ( x ) + by may a l s o be considered as a g e n e r a l i z a t i o n of AIZERMAN's [1] problem f o r n«2. Here f-^(x) and f 2 ( x ) are continuous f u n c t i o n s w i t h f-^(0) - f 2 ( 0 ) , and a and b are constants. We w i l l assume tha t a ^ 0, f o r i f a « 0 then v a r i a b l e s can be separated and the system can be i n t e g r a t e d immediately. Consider (1.14) x m cx + ay, y - dx + by where c and d are constants. I f R0UTH-HURWITZ c o n d i t i o n s c + b < 0 , cb ~ ad > 0 are s a t i s f i e d , then the zero s o l u t i o n x ( t ) « y ( t ) s 0 i s 2 7 . a s y m p t o t i c a l l y s t a b l e i n the sense of LYAPUNOV. Now the question a r i s e s : i f f l 1 5 ) f l < x ) f l ( x > f 2 ^ are f u l f i l l e d , then i s i t p o s s i b l e t o prove the asymptotic s t a b i l i t y i n the l a r g e f o r the system (1.1.5)? N . N. KRASOVSKII [ 5 ] showed th a t i f i n a d d i t i o n t o ( 1 . 1 5 ) we assume that ( 1 . 1 6 ) £ ( f 1 ( x ) b - f 2 ( x ) a ) d x - • as jxj -* «> x So' then the zero s o l u t i o n of the system ( 1 . 1 3 ) i s asymp-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 p e r t u r b a t i o n s . He f u r t h e r showed i n [ 4 ] that i f f x ( x ) f (x) f 2 ( x ) ( 1 . 1 7 ) ™ ~ - + b < - Y , - - ~ - - a . > Y , x + 0 where y > 0 i s a c e r t a i n constant, then the requirement ( 1 . 1 6 ) can be removed. We w i l l d i s c u s s the s t a b i l i t y of the system ( 1 , 1 3 ) by a q u a l i t a t i v e method i n order t o show that w i t h our assumptions (to be stated' y~ ':**"~" w''•"'*''' '* I'V*''•• the requirement ( 1 . 1 6 ) can be r e l a x e d . We prove the f o l l o w i n g theorem f p ( x ) f (x) Theorem 1 . 5 . Let h, (x) « —~-—, x 4° 0, h 0(x) - -=-r-r* J ^ 0 . i f ( 1 . 1 8 ) h (x) < 0, h (x) > 0 and a,b < 0 . w h e r e the f u n c t i o n s h^(x) and h 0 ( x ) are such t h a t the exis t e n c e and uniqueness of the s o l u t i o n s are guaranteed, then the zero s o l u t i o n of the .system. ( 1 . 1 3 ) i s asymptoti-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 p e r t u r b a t i o n s . Proof. We consider the curves given by xh^(x) + ay « 0 and xhg(x) + by • 0 . The or d i n a t e s of these, curves are r e s p e c t i v e l y as f o l l o w s 28. x h l ( x ) > z y l " a ~ 0 a c c o r d i n S as x ^  0 xh (x) > y 2 1 | 0 " " x ^  0 The curves and the axes d i v i d e the xy-plane i n t o e i g h t regions as shown i n F i g . 3 . x - x h 1 ( x ) + ay < 0 t o the r i g h t of the curve xh^xy+aysrO x - " " > 0 " " l e f t " " " " y . x h 2 ( x ) + by > 0 " " r i g h t " " 11 xh2(x)+by» 0 j m " » < o " " l e f t " " " " The p o s s i b l e d i r e c t i o n of a t r a j e c t o r y i s shown i n the F i g . 3 « We change to p o l a r coordinates and see th a t f and cp have the f o l l o w i n g signs I n the regions •  ( 1 , 5 ) r may be ^  QciaQd cp > 0 " " " (2,6) t < O.and cp may be t: 0 " " 1 1 ( 3 , 7 ) r may be > 0 and cp > 0 " " " (4 ,8) r < 0 and<cp may be > 0 The zero s o l u t i o n can be shown t o be a s y m p t o t i c a l l y s t a b l e i n the sense of LYAPUNOV. I n f a c t , l e t X p 2V 2 j^xh 2(x)dx - ay^ C l e a r 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 and vanishes only a t x • 0, y • 0. We compute d e r i v a t i v e of V w i t h respect t o time along an a r b i t r a r y t r a j e c t o r y of the system ( 1 . 1 3 ) • 2V « 2xh 2(x) x - 2ayy - 2 x h 2(x)[xh 1(x) + ay] - 2ay[xhg(x) + by] - 2 x 2 h 1 ( x ) h 2 ( x ) - 2aby2 < 0 f o r x =j» 0, y f 0. Consequently V i s a n e g a t i v e - d e f i n i t e f u n c t i o n . I t i s easy t o check, by a p p l y i n g the c r i t e r i o n of BENDIXSON (as ~ + - 2 i f 0 where X and Y denote the r i g h t hand sides 2 9 . FIG. 3 30. of the f i r s t and second equations r e s p e c t i v e l y i n ( 1 . 1 2 ) , t h a t there i s no p e r i o d i c s o l u t i o n . F i r s t we consider regions (2) , (4) , (6) and ( 8 ) . I n these regions f < 0; t h e r e f o r e any motion w i t h hounded p o l a r angle w i l l remain hounded i n each of these r e g i o n s . Second we consider regions (1) , (3) , (5) and (7) and show th a t there i s no motion w i t h hounded p o l a r angle i n any of these r e g i o n s , i . e . , we w i l l show th a t any motion s t a r t i n g I n Or e n t e r i n g any of these regions must leave t h a t r e g i o n a f t e r a s u f f i c i e n t time. We look a t - Y t-sincp. ( x h 1 ( x ) + ay) + coscp.(xh 2(x) + by) ] L e t us suppose th a t i n r e g i o n (1) there i s some motion which does not leave t h i s r e g i o n as t - °°. I n t h i s r e g i o n ^ < 0 , ^ > Q. Consequently two cases are p o s s i b l e . E i t h e r y i s bounded, i n which case the t r a -j e c t o r y remains bounded and t h e r e f o r e must have a w-l i m i t p o i n t d i f f e r e n t from ( 0 , 0 ) , which i s a c o n t r a d i c t i o n to our hypothesis; o r , y becomes i n f i n i t e l y l a r g e , i n urtiich case there e x i s t s a number k > 0 cp>_k> 0 which i m p l i e s t h a t cp does not remian bounded. Thus the t r a j e c t o r y must leave r e g i o n (1) and enter region (2) . A s i m i l a r reasoning holdjs f o r r e g i o n (5) . We consider the r a t e of change of the expression bx - ay along an a r b i t r a r y t r a j e c t o r y of the system (1 .13). ^•(bx - ay) - b ^ h ^ x ) + ay) - a ( x h g ( x ) + by) « x(bh,(x) - ah^(x) > 0 according as x ^ 0 . The s t r a i g h t l i n e bx - ay « A cuts the curve xh 1(x)+ay«0 f o r a l l A f 0 . I f A i s negative, then the l i n e bx-ayWl w i l l meet the curve xh^(x) + ay » 0 i n the f o u r t h quadrant. We examine the re g i o n bounded by the negative semi-y-axis, the l i n e bx - ay «• A and the curve xh^(x) + ay - 0 . Any motion s t a r t i n g i n or e n t e r i n g t h i s r e g i o n must cross the curve, since i t cannot cross the 3 1 . l i n e because along that t r a j e c t o r y ^r(bx-ay) > 0 . Moreover, t h i s motion cannot go t o the o r i g i n , as cp > 0 . A s i m i l a r reasoning w i l l hold f o r re g i o n ( 3 ) . For the l i n e L / 0 , » ) appearing i n ERUGIN's theorem, we can take the p o s i t i v e - s e m i - x - a x i s as a l l motions cut I t i n one d i r e c t i o n f o r a l l time while e n t e r i n g r e g i o n ( 1 ) from re g i o n ( 8 ) . Thus the hypotheses of ERUGIN's theorem are f u l -f i l l e d and the proof i s complete. As an example of the a p p l i c a t i o n of the above r e s u l t we may consider the f o l l o w i n g <systems example x - -xe~ x - 2 y y - x 5 - 3 y L e t t i n g h n ( x ) - -e~ x , a - - 2 , b - - 3 2 and hp(x) • x , we can e a s i l y check th a t the zero s o l -u t i o n of the above system i s a s y m p t o t i c a l l y s t a b l e I n the l a r g e . 1 . 5 . . The S t a b i l i t y , i n the l a r g e of the system ( 1 . 1 9 ) x - ax + f x ( y ) , y - f 2 ( x ) + cy where f ^ and f 2 are continuous f u n c t i o n s which s a t i s f y the u s u a l c o n d i t i o n s of existence and uniqueness and a and c are constants. We may add that t h i s system i s a l s o a g e n e r a l i z a t i o n of the problem of AIZERMAN [ l ] . This system was considered by N . N. KRASOVSKII [ 2 ] and I . H. MUFTI [ 1 ] . We prove the f o l l o w i n g theorem. Theorem 1 . 6 . L e t a < 0 , c < 0 , h..(y) < 0 and h 0 ( x ) > 0 f o r a l l x and y, and i n a d d i t i o n l e t | j v x h 2 ( x ) d x | -» » as |x| - co and _| y'yh 1(y)dy - -« as |y| ° ». Then the zero s o l u t i o n of the system ( 1 . 1 9 ) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . 32. Proof. We consider the f o l l o w i n g f u n c t i o n V • C x h„(x)dx - f yh,(y)dy o JO ; I t i s easy t o check that V > 0 and V = 0 only i f x = 0 and y = 0 . Computing the d e r i v a t i v e of V w i t h re s p c t t o time along a t r a j e c t o r y of the system (1.19) we have V = x h 2 ( x ) [ a x + y h x ( y ) ] - yh-^y) [xh 2 ( x ) + cy] = a x 2 h g ( x ) + x y h 2 ( x ) h 1 ( y ) - x y h 1 ( y ) h 2 ( x ) - cy 2h ]_(y) = a x 2 h 2 ( x ) - c y 2 h 1 ( y ) Obviously V i s a n e g a t i v e - d e f i n i t e f u n c t i o n , by v i r t u e of our assumptions, and i t vanishes only at x = 0 = y. Moreover V i s an i n f i n i t e l y l a r g e f u n c t i o n , i . e . , V•-» °° / 2 2 as A/x + y -• <». Consequently we can apply the theorem of E. A. BARBASHTN and N. N. KRASOVSKII and the proof i s complete. Remark. I . H. MUFTI [1] assumes: (1.20) a + c < 0 , ac - h 1 ( y ) h 2 ( x ) > 0 f o r x ={= 0 , y ± 0 and f1(0) = f 2 ( 0 ) = 0 and proved that " I f e i t h e r h x ( y ) > 0 , y £ 0, and h 2 ( x ) < 0 f o r x |=0, or h-^y) < 0 f o r y =f= 0 and h 2 ( x ) < 0 f or-x |= 0 , then the zero s o l u t i o n of the system (1.19) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e under c o n d i t i o n s (1 .20)." I f i n our assumptions we make the f o l l o w i n g change, namely, l e t h 2(x)>e>0 and h^(y)<y<0, then i t i s easy t o check t h a t the r e s u l t of the theorem remains tru e even i f we omit the c o n d i t i o n s that x ^ J x h 2 ( x ) d x -* <» and J y h-^(y) °°. Thus the assumptions made by I . H. MUFTI seem t o be i n c o r r e c t . We may p o i n t out that the r e s u l t of t h i s s e c t i o n cannot be dedueed from the r e s u l t s of the system (1 .3) . 3 3 -An a l t e r n a t i v e proof for,Theorem 1.2 Our f i r s t proof f o r Theorem 1.2 i s based on ERUGIN's theorem. In tha t proof we have shown that the requirement that • y G(yj = j hh 2(h)dh - » as |y|- 6 0 , can be rel a x e d . I t i s o p o s s i b l e t o prove Theorem 1.2 without u s i n g ERUGIN's theorem. This proof w i l l now be given. We r e c a l l that our assumptions f o r system (1 . 3 ) are (1.5) a < 0,e h ^ x ) <, c < 0,. (c f i n i t e ) , f o r a l l x ^  0 h 2 ( y ) > 0 and h-j(y). < 0 f o r a l l y ^  0 r y I t i s assumed th a t the f u n c t i o n G(y) <= J fth2(ft)dh, appearing i n the f u n c t i o n ' ° V = G(y) - \ a x 2 may not approach « as |y| •-• °°. I f we compute the d e r i v a t i v e of V w i t h respect t o t t , then by v i r t u e of system ( 1 . 3 ) we get V = y 2 h 2 ( y ) h 3 ( y ) - ax^h^x) . C l e a r l y V < 0 f o r x,y ^  0 and V(0,0) = 0 . ! We choose I > 0 and consider the set defined by V(x,y.) < I . I f t h i s set i s bounded f o r every I > 0, then we can use Theorem 0.8 to complete our proof, since the i n v a r i a n t set M = {(0,0,0)}. I f n o t , t h e n f o r any I > 0 and N > 0, the s e t G l , U = K x > y ) ; V < I, x > 0 , ^5±5 < y < - ^ J ij l ( x , y ) : V < I, o s v N ,r • N-ax -j i s bounded. I n order t o show t h a t the zero s o l u t i o n of system (1.3) 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 we s h a l l simply p r o v e t h a t a l l s o l u t i o n s of system (1.3) are bounded ( i n the f u t u r e ) . For, i f i t i s t r u e , then Theorem 0,8 can be used t o complete the proof. Let P ( x Q , y , z Q ) be an a r b i t r a r y p o i n t i n the phase space. Choose I and N • so.large t h a t p e G^  N» The t r a j e c t o r y cp(p,t), f o r t > 0"'(cp(p,0) = p ) , cannot leave G^ ^ without c r o s s i n g the boundary of G^  N- I t must cross e i t h e r V = I or one of'the four l i n e s y = + B and ax - cy = + N ( F i g . 4). Since V < 0* cp(p,t), f o r t > 0 cannot cross V = I. Now -grjr(ax-cy) = ax i|f(x) + ayhg(y) - cyhj(y ) < 0 f o r x < 0 and y > 0 > 0 f o r x > 0 and y < 0 = 0 f o r x = 0 = y , (where h ^ x ) = c + t ( x ) , \|r(x) jC 0). Therefore cp(p,t), f o r t > 0, cannot cross the l i n e s ax - cy = + N. I t may be n o t i c e d that 4^  < 0 and f i n i t e i n G. at every p o i n t on y - a x i s other than the o r i g i n . Therefore, from the nature of x, y and i t i s easy t o see t h a t cp(p,t), f o r t > 0 cannot leave. G^^ along or through any p o i n t on the y - a x i s . Furthermore y < 0 I n r e g i o n ( l ) and y > 0 i n r e g i o n (2) ( P i g , 4 ) . Co n s e q u e n t l y c p ( p , t ) , f o r t > 0 cannot c r o s s t h e l i n e s y = +B« The i n v a r i a n t s e t M, I n the p r e s e n t c a s e , i s {(0,0<,<))} . Hence, by Theorem 0 ,8 , .our p r o o f i s complete. Note t S i m i l a r p r o o f s can be g i v e n f o r Theorems 1.4 and 1.5. 36. Figure 4 STABILITY, IN THE LARGE OP -CERTAIN TYPES OP SYSTEMS OF THREE DIFFERENTIAL EQUATIONS 2 . 0 Theorems 0 . 4 and 0 . 6 r e q u i r e an i n f i n i t e l y l a r g e V - f u n c t i o n . This requirement, however, i s not always f u l f i l l e d , and consequent-l y the above mentioned theorems cannot be a p p l i e d . In case V i s not i n f i n i t e l y l a r g e (or V - 0 as x -• » ) , we can use L a S a l l e ' s ( [ 3 J , [ 4 ] ) Theorems on s t a b i l i t y f o r autonomous systems.. 1 Let V be a continuously d i f f e r e n t i a b l e f u n c t i o n on the Euclidean n-space R n t o the set of r e a l numbers R. I f G i s an a r b i t r a r y set i n R n, then V i s a LYAPUNOV f u n c t i o n on G f o r system . ( 0 . 8 ) ( i . e . , x = X(x)) i f V = (grad V)»X does not change si g n on G. Define E - {x : V(x) = 0 , x € Gi where "G i s the clo s u r e of G. Let M be the l a r g e s t i n v a r i a n t set i n E i . e . , a- set w i t h the property that a s o l u t i o n s t a r t i n g i n the set M remains i n s i d e i t f o r a l l t , p o s i t i v e or negative. M w i l l be a closed set. A fundamental s t a b i l i t y theorem [4], which i n c l u d e s a l l of the usual LYAPUNOV-like theorems on s t a b i l i t y and i n s t a b i l i t y , Is then the f o l l o w i n g : ' [ 4 ] : Paper on "An Invariance P r i n c i p l e In the Theory of S t a b i l i t y , presented at the Puerto R i c o Symposium, held i n December 1965. 38. Theorem 2.0; I f V i s a LYAPUNOV f u n c t i o n on G f o r ( 0 . 6 ) , then each s o l u t i o n x ( t ) of ( 0 . 8 ) that remains i n G f o r a l l t > 0 ( t < 0) approaches M* = M U l«J as t - » ( t -» -«). I f M i s hounded, then e i t h e r x ( t ) -» M or x ( t ) -» » as t . -» 88 ( t -• -oo). .Corollary: Let G he a component of Q n = {x : V(x) < hj . Assume that G i s bounded,. V _< 0 on G, and M° = U fl G c G. Then M° i s an a t t r a c t o r and G i s i n i t s r e g i o n of a t t r a c -t i o n . I f , i n a d d i t i o n , V Is constant on the boundary of M°, then M° i s a s t a b l e a t t r a c t o r . Note that i f M° c o n s i s t s of a s i n g l e p o i n t p, then p i s a s y m p t o t i c a l l y s t a b l e and G provides an estimate of i t s r e g i o n of asymptotic s t a b i l i t y . 2.1 S t a b i l i t y i n the l a r g e of a T h i r d Order Q u a s i - l i n e a r  D i f f e r e n t i a l Equation. Consider the d i f f e r e n t i a l equation ( 2 . 1 ) x* + ai' 1 ( x , x ) x + f^x^k + b f ^ ( x ) = 0 where a and b are some p o s i t i v e constants, and f-^(O) = 0. We assume that f u n c t i o n s which appear i n (2 .1) or i n any equation we s h a l l consider l a t e r , are continuous f o r a l l values of t h e i r arguments and s a t i s f y e xistence and uniqueness c o n d i t i o n s f o r the s o l u t i o n s . Any a d d i t i o n a l requirements w i l l be mentioned whenever and wherever needed. I t i s a l s o assumed that s o l u t i o n s of equations under c o n s i d e r a t i o n are defined on [0,8B) or (-»,»). 3y. ¥e w i l l t r y to f i n d some s u f f i c i e n t c o n d i t i o n s under which the zero s o l u t i o n of (2.1) 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 p e r t u r b a t i o n s . In order t o t e s t t h a t those s u f f i c i e n t c o n d i t i o n s do ensure the d e s i r e d s t a b i l i t y , a LYAPUNOV f u n c t i o n w i l l be constructed. Thus LYAPUNOV1s second method w i l l be our main t o o l . then we know that i t s zero 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 f a > b. However, i f a=b, then not every s o l u t i o n has the property that x ( t ) -» 0 as t -» ». We take a clue from t h i s I f we consider the equation x + a x + x + bx = 0, a > 0, b > 0 equation and assume that i' 1(x,x) , f' 2(x,x) and h^(x) = -^2-— (x ^ 0) taKe on p o s i t i v e values. A search f o r a d d i t i o n a l r e s t r i c t i o n s on f 1 ( x , x ) , f ^ ( x , x ) and f ^ ( x ) , w i l l be made i n order to ensure asymptotic s t a b i l i t y of the zero s o l u t i o n . The f o l l o w i n g n o t a t i o n s w i l l be used: y » y W(x,y) x y ab j f?(?)d§ +- b f 5 ( x ) y + j hg(x,h)dh g(x,y) f 2 ( ^ y ) - a J q a^Cx^nJdh , ax= |^  and y y j n a x g ( x , n ) d h I ( x , y ) Now we prove the f o l l o w i n g : 40. Theorem 2.1: I f f 2 ( x * y ) i s continuously d i f f e r e n t i a b l e , f ^ ( x , y ) twice continuously d i f f e r e n t i a b l e w i t h respect to x and ( i ) f 1 ( x , y ) 2 1* g(x,y) >. 1 f o r a l l x and y, 2 2 ( I : L ) | H < x f 5 ( x ) < 2- f o r a l l x , (2.2) ( i i i ) ag(x,y) - h f ^ ( x ) > p. 2 > 0 (pv > 0), f o r a l l x and y , ( ' = - ~ ) , ( i v ) y 3 x g(x,y) < 0 and | x j ( G ( x , y ) - l ) < .-|± </|l(x,y)| f o r a l l y ^ 0, • . —2 ' 2 (v) w(x,y) - » as „/x + y - », then the zero s o l u t i o n of (2.1) 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 p e r t u r b a t i o n s . Proof: Put ; y o z = y + a j f 1 ( x , h ) d n , then z = y + a f x ( x , y ) y + ax J S x f 1 ( x 3 h ) d h i y = -y f 2 ( x , y ) - b f 5 ( x ) + ay J 3 x f 1 ( x , h ) d h = -yg(x,y) - b f ^ ( x ) . Therefore (2.1) i s eq u i v a l e n t t o the system 41. (2.3) y = x = y z = z - a : f, (x,n )dn -yg(x,y) - b f 5 ( x ) Consider the f u n c t i o n V = i z 2 + w(x,y). I f we d i f f e r e n t i a t e V w i t h respect t o t , then hy v i r t u e of system (2.3) we get V = zz + a b f ^ ( x ) x + b f ^ ( x ) x y 4- b f ^ ( x ) y •+ yg(x,y)y + x J ha xg(x,h)dh = z[-yg(x,y) - b f 5 ( x ) ] + a b f ^ ( x ) y + b f 5 ( x ) y ' .y -y +bf 5(x).[z-a j f 1 ( x , h ) d h ] + y g ( x , y ) [ z - a J f- L(x,h)dh] .y + y I g(x , n ) d n ° = -y 2[ag(x,y)G(x,y) - b f ^ ( x ) ] - a b f ? ( x ) [ G ( x , y ) - l ] y + I ( x , y ) Obviously V = 0 f o r y = 0. However, i f y ^ 0, then we s h a l l consider two cases: Case 1: f 1 ( x , y ) = 1 . In t h i s case G(x,y) = 1 and V = - y 2 [ a f 2 ( x , y ) - b f ^ ( x ) ] + y J ^ c ^ f 2 ( x , h ) d h . Since a f 2 ( x , y ) - b f ^ ( x ) = ag(x,yj - b f ^ ( x ) 2 H > 0 and 4 2 . r y y 3 x f 2 ( x , y ) = y3 xg(x,y) < 0, which i m p l i e s that yj h d x f 2 ( x , y )dh<0, the r e f o r e V < 0 f o r y ^ 0. We may a l s o note th a t i f x = 0 , y ^ 0, then Y = -y 2[ag(0,y)G(0,y) - b f ^ O ) ] + 1(0,y) < 0 . Case 2; f 1 ( x , y ) 2 1. We may w r i t e V as Y = -y 2[ag(x,y)G(x,y) - b f ^ x ) ] - abf 5 (x.)y[ G(x,y)-1] , a 2 b 2 f ! ( x ) [ G ( x , y ) - l ] 2 a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ) 2 - i ! + l(x,y)+-£ 2 i — ag(x,y) - b f 5 ( x ) H a g ( x , y ) - b f ? ( x ) Let us look at U(x,y) = y 2[ag(x,y)G(x,y) - b f j ( x ) ] + a b f 5 ( x ) y [ G ( x , y ) - l ] 1 a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ] 2  7 [ a g ( x , y ) - b f j ( x ) ] t ' 2 Since ag(x,y)G(.x,y) - b f j ( x ) 2 ag(x,y) .- b f ^ ( x ) > p. > 0 , , , a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ] 2 and ^ [ a g ( x , y ) G ( x , y ) - b f ^ ( x ) ] ^ :  C a g ( x , y ) - b f 5 ( x ) ] - ^ a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ] 2 > 0 , x ^ 0, y ^ 0, the r e f o r e U(x,y) > 0 f o r a l l x £ 0, y ^ 0 . 43. I f we could show that a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ] 2 E(x,y) = I( x , y ) + 4 2 _ , < 0 f o r y ^ 0, ag(x,y) - b f 5 ( x ) then V < 0 f o r a l l y ^ 0. 2 . 2 . 2 , » , a b f , ( x ) p Now E(x,y).< I(xyy) + — _ [G(x,y) - 1] , y ^ 0 and th e r e f o r e E(x,y) < 0 f o r y ^ 0 provided t h a t -i-p- a 2 b 2 f 2 ( x ) [ G ( x , y ) - l ] 2 < - I(x,y) f o r Cy £ 0 (where by ( i v ) I ( x , y ) < 0 f o r y ^ 0), • • i . e . , i f a b | f 3 ( x ) | ( G ( x , y ) - l ) C < 2 j x y | l ( x , y j | , ,y ^ 0. By assumption ( i v ) |x|(G(x,y)-l) < f i . / | l ( x , y ) | , y ^ 0 and by assumption ( i i ) Therefore a b | f 5 ( x ) | ( G ( x , y ) - l ) . < a|x|(G(x,y)-l) < 2p./|l(x,y)| , 7 t 0, and consequently E(x,y) < 0, f o r a l l y ^ 0. Our next step i s to show that V i s p o s i t i v e - d e f i n i t e . For t h i s i t s u f f i c e s t o show th a t w(x,y) > 0 and w(0,0) = 0. We put r x r y F(x) = J f3(?)d§ and <£>(x,y) = J hg(x,h)dh 44. and w r i t e W.(x,y) = abF(x) + b f j ( x ) y + $ ( x , y ) [ 2 # ( x 3 y ) . + b y f ^ x ) ] 2 4abF(x) <£> (x,y) - y 2 b 2 f | ^$'(x,y) 4<£(x,y) f o r y / 0. Note that ^ ( x a y ) > 0, y ^ 0, w(0,y) =$(0,y) > 0 f o r y / 0 and f o r y = 0, x jL 0, w(x,0) = abF(x) > 0. . 2 2 2 Now 4 a b F ( x ) $ ( x , y ) - y b f-^( x) > 0 by assumptions ( i ) and ( i i ) . Therefore w(x,y) 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 :; r~2 ~2 Since w(x,y) -» » as Jx + y - », t h e r e f o r e V -* 0 0 as j—r> ^ P~ Jx + y + z - » and V(0,0,0) = 0. I t i s easy t o check t h a t the plane y = 0 i n which V vanishes does not c o n t a i n any e n t i r e t r a j e c t o r y of system (2 . 3 ) except the t r i v i a l one. Hence the proof i s complete by a p p l i c a t i o n of Theorem 0.6. Remark 1. Theorem 2.1 holds f o r the equation "x + a f 1 ( x ) x + f 2 ( x , x ) x + bf-j(x) = 0 w i t h G(x,y) and y3 xg(x,y) re p l a c e d by G(y) and y?> xf 2(x,y) r e s p e c t i v e l y . Remark 2. I f 1 > f (x,y) > e > 0, then Theorem 2.1 s t i l l holds i f we replace the c o n d i t i o n s ( i i i ) and ( i v ) by ( i i i ) ' ag(x,y)G(x,y) - bf' (x) _> p 2 > 0, and 45 . (iv)' jcxg(x,y) < 0 , |x| | G ( x , y ) - l | < -§H/I I ( x , y ) | y ^ 0 respectively. / 2.2 S t a b i l i t y i n the l a r g e of some T h i r d Order D i f f e r e n t i a l  Equations which are s p e c i a l cases of "x* + a f 1 ( x , x ) x .+... f 2(x,.x)x + bf-j(x) = 0 . Let us consider the s t a b i l i t y of the equation (2.4) "x" + f 1 ( x , x ) x + f 2 ( x , x ) x + bx = 0 i . e . , l e t a =. 1 and f-^(x) = x. Then assumptions (2.2), f o r equation (2.4), w i l l be replaced by r ( i ) f 1 ( x , y ) > 1 f o r a l l x and y 2 ( i i ) g(x,y) - .b 2 Ii > 0 f o r a 1 1 x and y ( i i i ) yS xg(x,y) < 0 and |x| ( G ( x , y ) - l ) < 2 l V | l ( x , y ) | f o r y ^ 0 ( i v ) w(x,y) - » as Jx + y - « . (we note t h a t g(x,y) > b guarantees t h a t w(x,y) > 0, x,y ^ 0) Or (*5)/f ^ ( i ) > f 1 ( x , y ) > € > 0 f o r a l l x and y 2 ( i i ) g(x,y)G(x,y) - b > p. > Cr f o r a l l N x and y ( H i ) y3 xg(x,y) < 0 and. | x | | G ( x , y ) - l | < 2|V | l(x,y) | y^O J~2 2" ( i v ) w(x,y) - » as J x + y -• » 46. (Note that assumptions ( i ) and ( i i ) imply that g(x,y) > b and consequently w(x,y) i s p o s i t i v e w i t h w(0,0) = 0). We may s t a t e the f o l l o w i n g : Theorem 2.2: I f i n a d d i t i o n to assumptions (2.5)' (or ( 2 . 5 ) " ) , 2 1 f ^ i s c and f 2 ( x , y ) i s c r e l a t i v e t o x, then the zero s o l u t i o n of equation (2.4) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . (To say that ^ i s c means that i t i s twice continuously d i f f e r e n t i a b l e ) We note that i n Theorems 2.1 and 2.2 we demand f 1 ( x , x ) t o be twice continuously d i f f e r e n t i a b l e w i t h respect t o x. I t w i l l be shown th a t i f i n equation (2.4) the above mentioned . ,. 2 1 requirement on f ^ ( x , x ) i s changed from c t o c then we can prove Theorem 2.3: I f f ^ ( x , x ) and f 2 ( x , x ) are continuously d i f f e r e n t i -able f o r a l l values of x, and ( I ) f 2 ( x , y ) > 1, f x ( x , y ) > b (b > 0), f o r a l l x and y (wi t h s t r i c t i n e q u a l i t y i n at l e a s t one), ( i i ) y a x(f 1(x,y.) + ^ f 2 ( x , y ) ) _< 0 f o r a l l values of x and y, then the zero s o l u t i o n of equation (2.4) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . Proof: Reduce equation (2.4) to an equivalent form (2.6) x = y y = z z = - f 1 ( x , y ) z - f 2 ( x , y ) y - bx, and consider the f o l l o w i n g p o s i t i v e - d e f i n i t e f u n c t i o n 4 7 . 2 2 y 2V = (z + by) + (bx + y) + 2bj n f- L(x , n ) d n o + 2 j n f 2 ( x,M)dn - (b*>l)y o I f we compute the d e r i v a t i v e of "V w i t h respect t o t , then by v i r t u e of system (2.6) we get 2V = - 2 { z 2 ( f 1 ( x , y ) - b) + b y 2 ( f 2 ( x , y ) - l ) ] - y y +2byj h S x f 1 ( x , h ) d h + 2yj h 3 x f 2 ( x , h ) d h = - 2 [ z 2 ( f 1 ( x , y ) - b) + b y 2 ( f 2 ( x , y ) - l ) ] + 2 b y J y h d x ( f 1 ( x , h ) + ^ f 2 ( x , h ) ) d h o Case 1: Let f-^x^y) = b and f 2 ( x , y ) > 1. Then V < 0 f o r y ^ 0 and V = 0 f o r y = 0. I f , i n a d d i t i o n , we assume that f 2 ( x , y ) J 1 as y -*,eo, i . e . i f f 2 ( x , y ) _> 1 + e, € > 0, then V i s a p o s i t i v e ^ d e f i n i t e i n f i n i t e l y l a r g e f u n c t i o n , • Since the plane y = 0 does not co n t a i n any e n t i r e t r a j e c t o r y of the system ( 2 . b ) we can use Theorem 0.b i n support of the a s s e r t i o n of our theorem. However i f f 2 ( x , y ) -• 1 as y -• • , then V i s not i n f i n i t e l y l a r g e . In f a c t , V-»0 as y-*03 along the l i n e z + by = 0 = bx + y. But the r e s u l t of our theorem s t i l l holds by c o r o l l a r y to Theorem 2.0, though i t i s not obvious. We w i l l give an indepen-dent proof i n a subsequent theorem where a s i m i l a r s i t u a t i o n occurs. 48. Case 2; f 2 ( x , y ) = 1 and f " 1 ( x , y ) > b. T h i s case can be t r e a t e d a l o n g s i m i l a r l i n e s as case 1. Case 3: f - ^ x ^ y ) > b , f g ( x , y ) > 1. I n t h i s case V < 0 f o r y ^ 0, z ^ 0, and V = 0 f o r y=0=z. S i n c e the x - a x i s does not c o n t a i n any e n t i r e t r a j e c t o r y o f system ( 2 . 6 ) , we can complete the p r o o f o f our theorem e i t h e r by u s i n g Theorem 0.6, i n case V i s i n f i n i t e l y l a r g e , or by C o r o l l a r y t o Theorem 2.0, i n case V i s not i n f i n i t e l y l a r g e ( i . e . , i n case f 1 ( x , y ) - b and f 2 ( x , y ) - 1 as y - » ) . I f i n e q u a t i o n (2.4) f 2 ( x , y ) i s not 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 , t h e n t h e r e s u l t o f Theorem 2.2 may n o t h o l d . I n t h a t case we may f o r m u l a t e Theorem 2.3*; I f f 1 ( x , y ) i s 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 w i t h r e s p e c t t o x and ( i ) f-L(x,y) > b + ^ ( f 2 ( x , y ) - l ) , b > 0 , f o r a l l x and y ( i i ) f 2 ( x , y ) _> 1 , f o r a l l x and y ( i i i ) y 9 x f 1 ( x , y ) _< 0 , f o r a l l x and y, t h e n the z e r o s o l u t i o n of e q u a t i o n (2.4) 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 . P r o o f : A f t e r r e d u c i n g (2.4) t o t h e f o r m ( 2 . 6 ) , we c o n s i d e r t h e f u n c t i o n ( p o s i t i v e - d e f i n i t e ) 2V = ( z + b y ) 2 + ( b x + y ) 2 + 2bJ "^vf x ( x , h )dh - b 2 y 2 49 Computing the d e r i v a t i v e of V w i t h respect t o t along an a r b i t r a r y t r a j e c t o r y of (2.6) we have 2V = 2(z + by)(z + by) + 2(bx + y)(bx + y) + 2 b y y f 1 ( x , y ) - T y s ' 2 • + i^bx h 5 f , ( x , h ) d h - 2b yy o x 1 = 2(z + by)(z + by + bx + y j + 2 b y z f 1 ( x , y ) J + 2by| h&i' (x,h )dh - 2byz J o x 'd(z + b y ) { - z f 1 ( x J ) y ) - f 2 ( x , y ) y - bx + bz + bx + y j + 2 b y z f 1 ( x , y ) + 2oyJ ha x± 1(x,h)dh - 2b yz - 2 { z 2 ( f 1 ( x , y ) - b) + y z ( f 2 ( x , y ) - l ) + by*(f.Jx 3 y) - l ) j •* y + 2by o hd f,dh The case fv,(x,y) = 1 has already been discussed In Theorem 2.2. Therefore we must take i" 2(x,y) > 1. C l e a r l y V = 0 f o r y = 0 , z = 0 , and i f y ^ 0, z ^ 0, then V < 0 only i f the matrix b ( f 2 ( x , y ) - l ) | ( f 2 ( x , y ) - l ) | ( f 2 ( x , y ) - l) f 1 ( x , y ) - b i s p o s i t i v e - d e f i n i t e , i . e . , i f f ^ ( x , y ) > b and b ( f 2 ( x , y ) - l ) ( f 1 ( x , y ) - b) > ±(f2{x,y) - i f or I f i" 1(x,y) > b + - ^ ( f 2 ( x , y ) - l ) , which i s t r u e by assumption ( i . ), The r e s t of the proof i s s i m i l a r t o t h a t of Theorem 2. 3, 50. For the equation ( 2 . 7 ) 'x' + f , ( x ) x + f p ( x ) x + bx = 0 we can de r i v e from Theorem 2.3 the f o l l o w i n g Theorem 2 . 4 : I i " h > 0 and ( i ) f 2 ( y ) > 1 f o r a l l y, y = x ( i i ) f x ( y ) > b + ^ j ( f 2 ( y ) - 1) then the zero s o l u t i o n of equation ( 2 . 7 ) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . Remark 1. We n o t i c e that i f i n equation (2 .7) f 2 ( x ) be replaced by f'2(x), then the r e s u l t of Theorem 2 .4 s t i l l holds (provided we repl a c e f 2 ( y ) by f 2 ( x ) i n assumptions ( i ) and ( i i ) a l s o ) . Remark 2. For asymptotic s t a b i l i t y i n the sense of LYAPUNOV of the zero s o l u t i o n of equation ( 2 . 7 ) we can simply assume b > 0, f 2 ( x ) > 0 and f - L ( x ) f 2 ( x ) > b. In order t o see t h i s , we reduce (2 .7) to the form x = y, y = z, z = - f 1 ( y ) z - f 2 ( y ) y - bx. Next choose an a r b i t r a r y f i x e d value y Q of y i n a neighbourhood of the o r i g i n , and form the c h a r a c t e r i s t i c equation 1 0 0 -X 1 f i ( y 0 ) - x 51-This equation w i l l have r o o t s w i t h negative r e a l p a r t s provided that c o n d i t i o n s f ^ ( y 0 ) > ®> f i ( y 0 ) D 1 f2 ( y 0 ) > 0 and h o l d , > 0 f i ( y G ) * ° i f 2 ( y 0 ) o o f ! ( y 0 ) b i . e . , b > 0, f-^yo) > 0 and f 1 ( y 0 ) f 2 ( y 0 ) > b. I f these c o n d i t i o n s are s a t i s f i e d , we can use Theorem 0.3 i n support of the a s s e r t i o n made above. In case 1 of Theorem 2.3, i t was promised to give an independent proof f o r the f a c t t h a t the s t a b i l i t y i n the l a r g e s t i l l holds when f 1 ( x , y ) - b and f 2 ( x , y ) •* 1 as y •* • . Without l o s s of g e n e r a l i t y , put b = 1 and consider the equation (2.8) x + x + f p ( x , x ) x + x = 0 Then we can s t a t e the f o l l o w i n g Theorem 2.4 ; I f fg ( x , y ) i s continuously d i f f e r e n t i a b l e w i t h respect to x and ( i ) f g ( x , y ) > 1 f o r a l l values of x and y (=k) ( i i ) y a x f 2 ( x ^ y ) _< 0 f o r a l l values of x and y then the zero s o l u t i o n of (2,8) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . 52. Proof: Replace (2.6") by an equivalent system x = y ( 2 . 9 ) y - z z = -z - f 2 ( x , y ) y - x and examine the f u n c t i o n 2V = ( z + y ) 2 + (x+y) 2 + (2 " n f P ( x , h ) d h - y c ) . 0 o I f we compute the d e r i v a t i v e of V w i t h respect t o t along an a r b i t r a r y t r a j e c t o r y of system ( 2 . 9 ) , then 2V = 2(z+y)(z+y) + 2(x+y)(x+y) y + 2 y y f 2 ( x , y ) + 2xJ ha xf 2(x,h)dH - 2yy = - 2 y ^ ( f 2 ( x , y ) - 1) + 2yJ n&xf2(x,h)dh . We n o t i c e that V < 0 f o r y ^  0 and V = 0 f o r y = 0. The f u n c t i o n V - 0 as y -» » along the l i n e z+y=0=x4-y, since we are assuming that f 2 ( x , y ) 1 as y -* *. Consequently, we cannot use Theorem 0.6. In order to prove the theorem, we consider the domain G^ N = {(x,y,z) : V(x,y,z) _< I , |y| < N], N > 0, l > 0. For |y| < N i t f o l l o w s from V(x,y,z) _< I that the x and z coordinates of the p o i n t s i n G^ N are bounded. Consequently G^ ^  i s bounded. Let P ( x 0 , y 0 , z 0 ) be a n , a r b i t r a r y p o i n t i n the phase-space. We denote by cp(p,t), f o r t > 0 the t r a j e c t o r y p a s s i n g through p and such that cp(p,0) = p. We choose I and N so that p l i e s i n G. ,T. Without l o s s of g e n e r a l i t y we can suppose: that the po i n t p i s not i n the plane y = 0, since a n y ' t r a j e c t o r y except the y . , . x . 2. 53. t r i v i a l one, s t a r t i n g i n t h i s plane does not remain i n i t f o r a l l t > 0. Therefore,, f o r P ( x 0 * y 0 * z 0 ) w e have V{ x0>yQ> z0) < I and |y | < N. We cl a i m that <p(p,t) e G. „ f o r a l l t > 0, i . e . . O 4s y IN ( i i i ) V ( c p(p,t)) < I, | y ( t ) | < N, f o r t > 0 . Suppose ( i i i ) f a i l s to hold f o r some t > 0 , then there i s a value T > 0 of t such that the po i n t cp(p,T) of cp(p,t) l i e s on the boundary of ( i i i ) . In a d d i t i o n , one or both of i n -e q u a l i t i e s i n ( i i i ) w i l l become e q u a l i t i e s . Since V ( c p(p,0)) < and V i s non-increasing along c p(p,t) f o r a l l t > 0, we cannot have the e q u a l i t y V ( cp(p,T)) = I. Suppose that the second i n e q u a l i t y becomes--'an e q u a l i t y ( i . e . , |y(T)| = N). In such a case we s h a l l d i s c u s s the signs of y(T) and Z(T). Now i Z(T) s a t i s f i e s the i n e q u a l i t y ( i v ) -N sgn y - H(x,N sgn y) < Z(T) _< -N sgn y + H(x, N sgn y) where H(x, N sgn y) = W21 - l ( x , N sgn y) - (x-hN sgn y ) 2 | / ' and the expression under the r a d i c a l s i g n takes on p o s i t i v e values not greater than 2l, and sgn y o I(x,N sgn y) = 2 hfp(x,h)dh - N . o eL For N s u i t a b l y l a r g e , we n o t i c e that i n ( i v ) f o r y = N, z i s negative and f o r y =• -N, i t i s p o s i t i v e . Hence y and z (z = y have opposite signs. Consequently, cp(p,t), f o r t > 0 , or any 54. ether t r a j e c t o r y that crosses the boundary of ( i i i ) must cross i t i n the d i r e c t i o n of decreasing y f o r y = N and i n the d i r e c -t i o n of i n c r e a s i n g y f o r y = -N. Thus, cp(p,t) f o r t > 0 remains i n s i d e G^ ^. Since p ( x 0 , y 0 , z 0 ) , I and N were a r b i t r a r y , we have shown that a l l s o l u t i o n s of system (2.9) are bounded ( i n the f u t u r e ) . Moreover, the only i n v a r i a n t set M i s {(0,0,0)}. Consequently, we complete the proof by a p p l i c a t i o n of Theorem 0,6. Remark. I f i n s t e a d of equation (2.8) we d i s c u s s the equation (2.10) x + ax + f 2 ( x , x ) x + bx = 0, a > 0, b > 0 under the assumptions ( i ) f g e c w i t h respect t o x ( ( i i ) f 2 ( x , x ) > -| f o r a l l values of x and x ( i i i ) y d x f g ( x , x ) X 0 f o r a l l values of x and x then we have the f o l l o w i n g i i Theorem 2.4 ; Under assumptions (2.11), the zero s o l u t i o n of equation (2.10) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . We may add t h a t a V - f u n c t i o n f o r the system x = y y = z, z = -az - f 2 ( x , y ) y - bx, which i s equivalent to equation(2.10), i s 2V = ab(z+ay) 2 + b 2 ( y + a x ) 2 + (2ab \ Y h f 9 ( x , h ) d h - b 2 y 2 ) J o * which g i v e s , by v i r t u e of the above system 55* 2V = -2aby"(af 2(x J )y) - b) + 2abyj h3 f 2 ( x , h ) d h . *~ o n I t may be pointed out that i f b < a then Theorem 2.4 t i n c l u d e s Theorem 2.4 . However, i f b > a then the r e s t r i c t i o n on f 9 ( x , y ) i n Theorem 2.4 i s stronger than t h a t i n Theorem 2.4 2.3 G e n e r a l i z a t i o n of AIZERMAN1s Problem i n the case when n = 5. V.A. PLISS [4] considered i n d e t a i l the f o l l o w i n g system of three d i f f e r e n t i a l equations: x = f (x) + a 1 2 y + a-^z ( 2 . 1 2 ) y = a 2 1 x 4- a 2 2 y +. a ^ z z = a^-j_x a^ 2y + a^^z ana discussed the s t a b i l i t y of the zero s o l u t i o n i n the l a r g e under v a r i o u s assumptions r e g a r d i n g the f u n c t i o n f ( x ) and the c o e f f i c i e n t s a ^ j . N. N. KRASOVSKII [ l ] considered the f o l l o w i n g more general case x = f 1 ( x ) + a 1 2 y + a-^z ( 2 . 1 3 ) y =-f2(x) + a 2 2 y + a ^ z z = fj(x) 4- a^ 2y + a ^ z where a ^ are a r b i t r a r y constants, f ^ x ) are continuous f u n c t i o n s w i t h f^(0) = 0 and which s a t i s f y c o n d i t i o n s of e x i s -tence and uniqueness of the s o l u t i o n s . He f u r t h e r assumed that 5b. 2 2 2 A l l + A 2 i + A 3 i ^  °> w h e r e A n = a 2 2 a 3 3 " a 3 3 a 3 2 > A2^ = a]_^a^2 ~ a l 2 a 3 3 and. ~ a l 2 a 2 3 ~ a ^^ a 22 " N. N. KRASOVSKII proved v a r i o u s r e s u l t s f o r (2.13), one of which f o l l o w s . Let a 1 2 A 3 1 = SL.^-^2l a n d t h e c o e f f i c l e n t s of the equation x " 1 f 1 ( x ) - X x - 1 f 2 ( x ) x _ 1 f , ( x ) a 12 a 2 2 " X a 13 a a 32 23 a 3 3 " X = X^ + a ( x ) X 2 + b(x)X + c(x) = 0 f o r a l l x ^ 0 s a t i s f y the i n e q u a l i t i e s a(x) > 0, a ( x ) b ( x ) - c ( x ) > c(x) > 0, (analogous t o c o n d i t i o n s of ROUTH-HURWITZ i n the case of a l i n e a r system), then i n order that the zero s o l u t i o n of the system (2.17) he a s y m p t o t i c a l l y s t a b l e In the l a r g e , i t . i s necessary and s u f f i c i e n t that l i m L21 [ x ( - a ( x ) ) + s i g n x - xc(x)dx] = - « x - +o= a l 2 J .x We w i l l f i r s t of a l l , consider the f o l l o w i n g system of three d i f f e r e n t i a l equations, namely, (2.14) x = (j)^(x) + az y = <|>2(y) + bz z" = (^(x) + <p4(y) + cz 5 / . I t w i l l be shown tha t we can use i t t o derive some r e s u l t s f o r the system x = f 1 ( x ) + a 1 2 y + a-^z (2.15) y = a 2 ]_x + f 2 ( y ) + a 2^z z = a ^ x + a^ 2y + a-^z which may he considered a g e n e r a l i z a t i o n of AIZERMAN's problem i n the case when n = 3-In (2.14) we assume that ^ ( 0 ) = <P2(0) = <j)^ (0) = <P4(0) = 0 and ( i ) a > 0, b > • 0 and c < 0 (2.1b) ( i i ) x(j) 1(x) _< 0, x(p 5(x) < 0, f o r a l l x ji 0 ( i i i ) y<|)2(y) < 0, y(() 4(y) < 0, f o r a l l y ^ 0 (with s t r i c t i n e q u a l i t y e i t h e r i n f i r s t member of ( i i ) or that of ( i i i ) ) . The f o l l o w i n g theorem can now be proved Theorem 2.5; I f , i n a d d i t i o n t o assumptions (2.1b), i-x r y M <M§)d?| - • as |x| - • and |J <j>4(h )dh | - » o o as |y| - «, then the zero s o l u t i o n of (2.14) i s asymp-t o t i c a l l y s t a b l e i n the l a r g e . Proof: Consider the f u n c t i o n P x _ y V = -gz - - < M § ) d § - ^ j <|)4(h)dn o ^ o 5«. In view of assumptions ( 2 . l 6 ) , V i s a p o s i t i v e - d e f i n i t e i n f i n i t e l y l a r g e f u n c t i o n . I f we compute the d e r i v a t i o n of V w i t h respect to t , then hy v i r t u e of system (2.14), we get V = zz - A <|>5(x)x - ^  <j)4(y)y = z[(J)3(x) + <{)4(y) + cz] - -| <t)5(x)[(J)1(x) + azj - -5 i\(y ) [ < P 2(y) + bz] = c z 2 - \ ^ ( x ) «j)3(x) - £ <P2(y) <p4(y) C l e a r l y V < 0 f o r z ^  0 and V = 0 at the o r i g i n and p o s s i b l y i n the plane z = 0. I t i s easy t o v e r i f y that the plane z = 0 cannot contain any e n t i r e t r a j e c t o r y of system (2.14). Consequently, we can complete the proof by a p p l i c a t i o n of Theorem 0.6. Note. The requirement t h a t a and b are non-negative cannot be removed unl e s s , of course, we Introduce some a d d i t o n a l r e s t r i c -t i o n s on (p^'s. In order to see t h i s , l e t us r e w r i t e (2.14) i n the form <Pi(x) x = — - — . x + 0 . y + az <p2(y) y = 0 • x + — y — • y 4- bz <Mx) <j)4(y) z = . x + . y + cz and form i t s c h a r a c t e r i s t i c equation, namely the equation - X 0 X 3 2 , M x ) M y ) X - X [ -i=:— + x y o <P2(y) — x y <P4(y) y + c] + x[ c - x ^(x) (j)2(y) x + c ( - i - — + —) + b \ r + a' ^ <l>i. (y) <MX) y X ] y _ ^OO $ 4 ( y ) <|)2(y) Mx) ^(x) (j) 2(y) , + [b • • + a • —— - c • L x y y x - x y J 0 f o r x,y ^  0 . In view of assumptions ( i i ) and ( i i i ) i n (2.16), i t i s necessary t h a t a and b be non-negative f o r the above equation t o have r o o t s w i t h negative r e a l p a r t s . We now dis c u s s system ( 2 . 1 5 ) . Case 1: a ^ 2 or a 9 1 ^ 0, and a.OA, a n ^  ^  0. Consider the f o l l o w i n g "21 2 3 ' "13 l i n e a r non-singular t r a n s f o r m a t i o n xn = X - , = y a _21 L23 x + -12 l 13 y + z then 6o. x l ( fl<*l> ' x l ) + ^ a 1 3 Z l (2.i7) y x - ( f 2 ( y i ) - Y i ) + a 2 3 z i 13 2 "XT •* a 3 3 a 2 1 51 1 a23 2 , a l 2 „ 9* -t o 3* r\ ~2L Xd y l 13 1 a13 a o n a. I f we Introduce the n o t a t i o n s 2 a21 = f j l g c = a - a53a21 _ a 2l a 13 a23 ^ a13 ' 5 1 a23 a23 2 2 2 a33 a l2 a i2 a 23 a21a13 + a12a23 d = a , 0 - - ~ — , e = a,, + — = 32 a 1 5 a 1 ? 33 a i3 a 23 then system (2.17) assumes the form x = ( f x ( x ) - a a 1 5 x ) + a l 5 z (2.17* ) y = ( f 2 ( y ) - ^ a 2 5 y ) + a 23 z z* = ( a f 1 ( x ) + cx) + ( h f 2 ( y ) + dy) f ez . I f ( i ) a 1 j , a 2 3 > 0, e < 0 (2.18) ( I i ) x ( f 1 ( x ) - a a 1 3 x ) < 0, x ( a f 1 ( x ) + cx) < 0, x ^  0 ( i i i ) y ( f 2 ( y ) - h a 2 ? y ) < 0, y ( b f 2 ( y ) + dy) < 0, y ^  0 ( with s t r i c t i n e q u a l i t y e i t h e r im, ( i i ) or i n ( i i i ) ) then the f o l l o w i n g theorem can be sta t e d : 61. Pheorem 2.6: I f , In a d d i t i o n to the assumptions ( 2 . l 8 ) , r x | ( a f , ( ? ) + c?)d?| - • as |x| -» • and J o 1 IJ ( b f 2 ( n ) + dh)dh| - » as |y| - » , then the zero s o l u t i o n of system (2.17') 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 p e r t u r b a t i o n s . Case 2t aip_ — a21 — ^ In such a case, system (2.15) assumes the form -3Z x = f-j_(x) + a-j^. (2.19) y = f 2 ( y ) + a 2 3 z z = a ^ x + a^ 2y + a-^ -^ z I n what f o l l o w s i t w i l l be assumed t h a t a 2 ^ , a ^ 2 ^ 0. Consider the f o l l o w i n g f u n c t i o n 1 2 1 2 1 2 V = ^ a 2 3 z - -g-a^y + ^ x . Obviously, V i s p o s i t i v e - d e f i n i t e i f a ^ > 0 and a ^ 2 < 0. Note th a t t h i s assumption does not c o n t r a d i c t assumptions (2 . l 8 ) Therefore, i f we suppose that t h i s i s t r u e , then V - • as 2 2 2 r = Jx + y '+ z ~* 0 8 . I f we compute the d e r i v a t i v e of V w i t h respect t o t , then by v i r t u e of system (2.19) V = a ^ z z - a^ 2yy + xx = a 2 3 z [ a 3 1 x + a^ 2y + a ^ z ] - a 5 2 y [ f 2 ( y ) + a ^ z ] + x [ f x ( x ) + a 1 5 z ] 2 = x f x ( x ) + a ^ a ^ 2 - z x ( a 2 5 a 5 1 + a ^ ) - x f 1 ( x ) ] - a ^ 2 y f 2 ( y ) . 6 2 . I f we f u r t h e r assume th a t y f g ( y ) _< 0 f o r a l l y, x f 1 ( x ) < 0 for x j£ 0, a.-,-, < 0 and 1 2 2 a 2 ^ a ^ x f 1 ( x ) - -jpx ( a ^ a - ^ + a ^ ) > 0, x ^ 0 , then V < 0 f o r x,z ^ 0 and V = 0 at the o r i g i n and p o s s i b l y on the l i n e x = 0 = z. Since the y-axis does not co n t a i n any e n t i r e t r a j e c t o r y of system (2.19) f o r a l l t > 0, except the t r i v i a l one, then by a p p l i c a t i o n of Theorem 2.6 we a r r i v e at the f o l l o w i n g Theorem 2.7: I f &2 > 0, a ^ , a ^ 2 < 0, ( i ) x f 1 ( x ) < 0 f o r a l l x ^ 0 ( i i ) y f 2 ( y ) < 0 f o r a l l y 2 2 ( i i i ) 4 a 2 5 a 3 5 x f 1 ( x ) > x ( a ^ a ^ + a-^) , x £ 0 then the zero s o l u t i o n of system (2.19) i s a s y m p t o t i c a l l y s t a b l e i n the l a r g e . 2.4 S t a b i l i t y i n the l a r g e of system (2.20) Consider x = f-^x) + a l 2 y + a 1 5 z (2.20) y = f 2 ( x ) + a 2 2 y + a ^ z z = a ^ x + a^ 2y + f ^ ( z ) where f . ( 0 ) = f p ( 0 ) = f , ( 0 ) , and f-,(x), f p ( x ) and f , ( z ) are 63. continuous and s a t i s f y other c o n d i t i o n s which guarantee the existence and uniqueness of the s o l u t i o n s of the above system, i n the e n t i r e phase-space and a j _ j ' s a r e constants. We w i l l assume that f ^ ( z ) i s not n e c e s s a r i l y a l i n e a r f u n c t i o n of z, since i n that case our system i s a p a r t i c u l a r case of the system considered by KRASOVSKII [ l ] . We suppose f i r s t of a l l , that a l 2 , a ^ 2 t 0 and at l e a s t one out of a.^ and i s not zero. We may perofrm the f o l l o w i n g l i n e a r non-singular t r a n s f o r m a t i o n : x.^  = x a 3 1 a i 3 ? 1 = a x + y + — ^ z x a ^ 2 a l 2 z^ = z and get a- a i x = f ^ ) + a 1 2 [ y i - - j S X l . _ i 5 Z l ] + a i j Z l = < W - xi» + a i 2 y i z ' i = a ? i x i + a ? 2 [ y i - x l ' z l ] +" f 3 ( z i ) " a J 2 y l + - Z l » a31 - . a!3 -yn = a x + y + a z i 3 2 a 1 2 - 5 1 [ f 1 ( x 1 ) - , 1 2 3 1 X l + a 1 2 y 1 ] + f 2 ( x 1 ) a^g j . • j . -. c , 3 2 + a 2 2 [ y i - i | x i - z i ^ + 1^  f ^ i + f 3 ( z i } _ ^ 3 2 ! l 3 ] a 1 2 1 64. 2 = f l ( x l ) + f 2 ( x l ) ' x l ( + ^ | ^ ) ] a J 2 1 - 1 1 a ^ 2 a32 + y [a + a ^ 1 & 1 2 + a 1 3 a 3 2 1 22 a 5 2 a ^ 2 a12 * 1 1 a i 2 a ^ 2 Suppressing the s u f f i x 1 from the v a r i a b l e s we assume the s y s t e m i n the form x = ^ ( x ) + a 1 2 y (2.21) y = (J)2(x) - by + ^ ( z ) z = a^ 2y - 4>i,.(z) where v ( M x ) = f , ( x ) - i | ! 2 1 x J> a l 2 ^ a l 2 a i 2 (j) 4(z) = - f 3 ( z ) 4- -2§!i5 z > - ^ 2 2 - ^ f ) and ^ ( 0 ) = <J)2(0) = <J)5(0) = (|)4(0) = 0 . Let us assume th a t 1 ( i ) b > 0, a 1 2 < 0, x(() 1(x), x(j) 2(x) > 0, x ^ 0 (2.22) ( i i ) z(j) 4(z) > 0 f o r a l l z 0 ( i i i ) 4bz(]) 4(z) j> ( z a ^ 2 + ^ ( z ) ) 2 f o r a l l z 6 5 . a n d prove the f o l l o w i n g ; . Theorem 2.8; I f i n a d d i t i o n to assumptions (2.22), x o ^ ( ^ •"* 0 0 as jxj -» » , then the zero s o l u t i o n of system (2.21) 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 p e r t u r b a t i o n s . Proof: Consider the f o l l o w i n g f u n c t i o n 1 , 2 . 1.2 l _ j X M ? ) d ; ld O v - *y + ¥ a. I f we compute the d e r i v a t i v e of V w i t h respect t o t along an a r b i t r a r y t r a j e c t o r y of system (2.21), then V = yy •+ zz - \— <{>p(x)x 12 ' = y[<P 2( x) ~ b y +' ^ C 2 ' ) ] + z [ a 5 2 y - <j) 4(z ) j V i - <P2(x)[<t)1(x) + a l 2 y ] = - "S^ -J .:<t>1(x)<|>2(3c) - [ b y 2 - y(<j) 5(z) + a 3 2 z ) -r- " + Z<j) 4( Z)]. Let U(y,z) = by2'-- y(<t>3(z) + a-^z) + z(p4(z) Note that U w i l l be always non-negative i f b > 0, z(p4(z) > 0 f o r a l l z.^ 0 and 4bz(j)^(z) _> ((j)-j(z) + a^ 2z) f o r a l l z. These c o n d i t i o n s . a r e s a t i s f i e d by assumptions (2.22). Consequently, we conclude t h a t 66. V < 0 . f o r x ^ 0 and = 0 at the o r i g i n and p o s s i b l y i n the plane x = 0. I t i s easy t o v e r i f y that the plane x = 0 does not contain any e n t i r e t r a j e c t o r y of system (2.21) f o r t > 0, except the t r i v i a l one. Hence we use Theorem 0.6 to complete our proof. CHAPTER I I I ON A PERIODIC SOLUTION OF A HAMILTONIAN SYSTEM WITH n DEGREES OF FREEDOM 3 . 1 . I n t r o d u c t i o n . (3.1) This chapter extends the r e s u l t of C. L. SIEGEL [1], f o r c o n s t r u c t i n g a p e r i o d i c s o l u t i o n f o r a Hamiltonian system w i t h n degrees of freedom, i n a sense t h a t w i l l he made c l e a r i n the sequel. We consider the Hamiltonian system -H_ 1,2, n. where H i s a power s e r i e s i n x^,y^. (k == 1,2,..., n) without a constant term and w i t h r e a l c o e f f i c i e n t s . , Furthermore, we assume that the o r i g i n i s an e q u i l i b r i u m p o s i t i o n so th a t a l l H , H vanish at the o r i g i n and the x f y k power s e r i e s f o r H begins w i t h quadratic terms i n x f c, y k (k = 1,2,..., n ) . I f we put (3.2) Z k " X k k < n k > n then we have 1 Ul Ml n H » 4 E S /L z,z + where S = ( Z ^ ) i s a r e a l 2n x 2n matrix L e t H z -• x i . n H and J = r 0 E -E 0 _ 68 (3.3) where E i s a n x n i d e n t i t y m a t r i x . 1 Then we may put (3.1) i n the form z = JH or z L e t t i n g A » JS we have z = AZ + I t i s well-known th a t the c h a r a c t e r i s t i c equation of the system (3.3) has the form c p ( X ) o | X E - A| = X 2 n + b' 1X 2 n"' 2 + ... + b n = 0 where E i s now 2n x 2n i d e n t i t y m a t r i x . I f X i s a root of c p ( X ) = 0 , then so i s - X . Now arrange the roo t s X ^ i - X g , . . . , X g n of c p ( X ) = 0 i n t o the two groups. ^l 3 ^2* " * *' ^ n' ~^ 2.* '' °•* ~i^n* Since the c o e f f i c i e n t s of cp(X) = 0 are r e a l , i t f o l l o w s t h a t 3 ^ I s an eigenvalue ( i . e . the root of cp(X)= 0) I f X k i s . I f X f e f a i l s t o be r e a l we have the two cases: (1) X k i s p u r e l y imaginary or ( i i ) X k has a non-zero r e a l p a r t . I n the f i r s t case X ^ = -X k = ~Xk, we s h a l l assume t h a t p u r e l y imaginary r o o t s are grouped together i n two groups mentioned above. Suppose X k =» a + i b w i t h ab ^ 0 , then "X~k = a - i b i s a l s o an eigenvalue and i f necessary we s h a l l r e l a b e l i t as "k^l < n)/ Then X k + n = - X k = -a « i b and X = -X. = -a + IbvVfe w i l l assume th a t these eigenvalues i . e . X k and X ^ belong t o the same group. 69 I f X k i s r e a l so i s -Xfc = \ + n ' Consequently we can assume tha t X^ > 0. Consider now \theoequation ( 3 . 4 ) t => JH_ = Az + ..., 2 JS I t was shown hy C. L. SIEGEL [ 1 ] t h a t there e x i s t s a non-singular constant matrix C 3 A C • - where D n -X-, I f we make the s u b s t i t u t i o n (3.5) z = Cz< -X n then ( 3 - 4 ) becomes . - 1 ( 3 . 6 ) z* = C" xACz* + ... = Dz* + I t i s known tha t C can be determined only upto an a r b i t r a r y non-singular diagonal matrix f a c t o r D* and that D£ can be determined i n such a way that t r a n s f o r -mation (3.5) i s c a n o n i c a l . I f we w r i t e (3 .5) as C l z ! + + G 2 n 2 2 n s i n c e , X k i s an eigenvalue and i t s corresponding eigenvector, we can show tha t C^= C^ provided X^ = X f e. Hence the necessary and s u f f i c i e n t c o n d i t i o n f o r z to be r e a l i s that 70 z*:« z£, where k and i are d i s t i n c t chosen from 1,2,..., n or from n+1, .. .., 2n. Now we put z k x k zK+n = ^ k k 1 * 2 , . . . , n . I f the o r i g i n a l v a r i a b l e s n a r e r e a l and the tr a n s f o r m a t i o n (3.5) i s t o he c a n o n i c a l , then, as shown hy C. L. SIEBEL [1], the f o l l o w i n g cases occur Case 1. The v a r i a b l e s x£, y* are r e a l i f Xfc i s r e a l . Case 2. I f Xfc = a + i b , ab ± 0 and X^ = "X"k ( l _< k, I _< n) then Case 3. I f X k i s p u r e l y imaginary, then x£ - i y j k » 1,2,...., n. y k = i x k 3.2. C o n s t r u c t i o n of I a. P e r i o d i c S o l u t i o n Consider the Hamiltonian system w i t h h(n _> 2' ) degrees of freedom. 1 We suppose that eigenvalues of the matrix of the corresponding l i n e a r system are arranged according t o the d i s c u s s i o n of the s e c t i o n 3.1 ( i . e . as X^,X 2,...,X ni -X^,-X2,,..-X . ) . L e t us assume the c o n d i t i o n s : (a) X^, X m, m |= 1, n are p u r e l y imaginary. ("b) X k, i s of the form a + i b , ab ^ 0 f o r k f 1, m and 1 < k _< n. ( i f n=2 then m=2 and k=0 ) 71 (c) \r>3 ..., \ • are l i n e a r l y independent over the r a t i o n a l s , and (d) the system i s already i n the normal form, namely (3.7) . z k = \ \ + f k ( z ) , k = 1,2, . . . , 2 n where f k ( z ) are power s e r i e s i n z ^ , z 2 , . . . , z 2 n beginning w i t h quadratic terms. The r e a l i t y c o n d i t i o n s f o r the three eases depending on the nature of \ k were p r e v i o u s l y given f o r the t r a n s f o r m a t i o n . z* = Cz v For convenience we omit the s t a r on the c o n d i t i o n s , given i n sec. 3.1. We s h a l l look f o r a s o l u t i o n of (3.7) i n - t h e form CO (3.8) z k = £ c p j ^ , k = 1 , 2 , . . . , 2n where c p ^ . ^ i s a homogeneous polynomial of degree i i n the new time dependent v a r a i b l e s a, P, Y and &• Now, l e t us make the f o l l o w i n g assumptions: I The variables^ a, p , y and 6 s a t i s f y the d i f f e r e n t i a l equations: d = aa, $ = bp (3.9) and y = CY , o = d 6 where a =i a Q + a^(aPY6) + ag(aPY6) + ^ ^ b = b Q + b 4 ( a p Y6) + D 8 ( a P Y 6 ) 2 + ........ c = c o + C2,(apY6) + c 8 ( a p Y 6 ) 2 + .. .. d = d Q + d^(apY6) + d g ( a P Y 6 ) 2 + 72 I I f o r k=l • " k=m " k=n+| " 'k=n+m otherwise I I I mJ Jn+1 and z n+m do not contain any term of the form a P P q y r o 3 P = 'r+1 and q=S j q = s+1 and p = r ; r = p+1 and q = s; and p = r and 5 = q+1 r e s p e c t i v e l y , where p,q, r and S are p o s i t i v e i n t e g e r s . tkA*, I f we s u b s t i t u t e (3-8) i n t o (3.7) and re-order them, i n the form % - X xz f c = fk(zK we have ( 3 . H ) r 00 " = f k where c p k r r ^ i s p a r t i a l d e r i v a t i v e of. c p k r w i t h respect t o ^ ( £ = ' a , P , y and 6 ) . We proceed t o equate the c o e f f i c i e n t s of the terms of degree l i n a , P , y a n ^ & o n both sides of (3.11)• For 1=1 there i s no c o n t r i b u t i o n from f ^ since they s t a r t w i t h terms of second degree. By (3.8) z^ has no constant term, t h e r e f o r e we get i f k - n+1 i f k = n+m h i f k (3.12") c o = Xn+1 (3.13*) b = o X • m i f k (3.13") d = o Xn+m = ~K m For i > 1 we make, the i n d u c t i v e assumption.) th a t cp^ are known polynomials f o r r < t3 k = 1 , 2 , . . . , 2n. 73 Then a ,b ,c and d are known constants f o r r <. l - l , • * I* I* I* I* • i hy 4. where a.,b.,c. and d. are a l l z e r o . i f i i s not d i v i s i b l e For f i x e d k and t l e t i n e p ^ the term of the, form a P p q Y r 6 S , where p + q + r + s = ibe GaP^yrtB . Since the f ^ ' s s t a r t w i t h quadratic terms, the, c o e f f i c i e n t of - a p p q y r 6 S ; on the r i g h t hand side of (3.. 11) i s known by i n d u c t i v e assumption, c a l l t hat c o e f f i c i e n t S^. On the l e f t hand side of (3.11) the c o e f f i c i e n t ' of a" p j, a p { 3 q Y r 6 S i s (3.14) a opC + b QqC + c Q r C + d QsC - XfcC + Rfc, where R k i n v o l v e s a^, b^, c^ and d^ f o r i _< I - 1 and c o e f f i c i e n t s of c p ^ f o r r _< l-l. I f k does not take any of the values l,m,n+l, and n+m, then cpj^ =0. Consequently a^, b^, c^ and d^ occur I n Rfe only f o r i _< l and are known q u a n t i t i e s by i n d u c t i v e assumption. I n order t o determine C (and hence c p k r f o r a l l p o s i t i v e i n t e g e r r) by means of ( 3 . 1 5 ) C(a Qp + b Q q + c Q r + a Qs - Xfc) + R r = Sfe i t reamins ( i ) to determine C when a p + b a + c r + d s - X, = 0 v 1 o o o o k and ( i i ) t o ob t a i n a.^  ^ , b^ ^ , c^ ^  and i n terms of a i , b i , c i and d^ f o r i < 1-2 and the c o e f f i c i e n t s of cp f c r f o r r _< i - 1. To t r e a t ( 1 ) we note that i n view of (3.12) and (3.13) we have (*) a o p + V + c o r + V - \ = x i ( p " r ) + \aCq-8) - \ I f k f - 1 , m, n+l and n+m, then the above expression can be zero only I f the r e a l p a r t of \ k i s zero, which i s a 74 c o n t r a d i c t i o n since X k = a + i b , ab ± 0. I f k = 1 , then (*) becomes X 1(p - r - 1 ) + X m ( q - s) which i n view of the l i n e a r independence of X-^ , Xg, i s zero only i f p = r + 1 and q = s. But i n t h i s case assumption III i s c o n t r a d i c t e d . Thus i n t h i s case C = < The reasoning f o r the cases when k = m, n + 1 or n+m i s s i m i l a r . But t h i s c o n d i t i o n (G = 0) i s p r e c i s e l y the s i t u a t i o n which we can use to determine a^, b^, c^ and d. f o r i <_ i-l. I f k=d, then p = r + 1 and q = s i t then f o l l o w s t h a t ( 3 . 1 5 ) becomes R l - S l and i n (3.11) f o r k=L, a .. occurs as the c o e f f i c i e n t of a(ay) -(£6) where u i s a p o s i t i v e I n t e g e r , only i n the term aacp-Qa and t h e r e f o r e the term c o n t a i n i n g i t i s l-l , u a ,a(a0y6) 4 . Comparing the c o e f f i c i e n t s of d(apyS) Jf - 1 w i t h u = y i e l d s a^ ^  i n terms of a^, b^, c^, d^ f o r i _< 1-2 and the c o e f f i q i e n t s of f^. and cp k r f o r r <_ £ - 1 , a l l of which are known. .So a , i s uniquely determined S i m i l a r l y , b^_^,c^_-^, d a r e uniquely determined. Suppose we are able to prove convergence and, moreover, ( 3 . 1 6 ) a + c = 0= b + d Then we get from ( 3 - 9 ) U O (dy) = 0 = (p6) i . e . , ctY = a v . P6 = £,6 or 0 0 o o ( 3 . 1 7 ) apy6 = a 0 P 0 Y Q 6 o , where a Q , 0 o , Y o and §,Q are the i n i t i a l values of OS,]3,Y and 6 r e s p e c t i v e l y . I n view of 75 (3-9) t h i s i m p l i e s that a,b,c and d are constants. I n t e g r a t i n g (3-9) we get (3.18) a = a Q e a t , y =-y oe~ a t 0 = 6 e b t , 6 = 6 e" 1^ o o Now, i f we can prove that a and b are pu r e l y imaginary we w i l l have a p e r i o d i c s o l u t i o n . To t h i s end we make use of the Hamiltonian character of (3-7). The Hamiltonian system i s (3.19) * k = Hy , y k = -H , k=l , 2 ,...,n. where H i s independent of t . So m u l t i p l y i n g the f i r s t equation i n (3.19) by y^, the second by -x^ and adding the two and then summing over k from 1 t o n we o b t a i n (3.20) K | ( V K + V K ) = S = ° i . e . H = constant i s an i n t e g r a l of ( 3 . 1 9 ) . Suppose now that we s u b s t i t u t e (3-8) i n H (keeping (3-9) i n mind), then from (3.20) we o b t a i n (3-21) H = H aa + HQb|3 + H cy + H.d6 = 0 a p • , y o ; i n d e n t ! c a l l y i n a,0,y a n (3 6. By (3-8) and assumption I I I , we have (3-22) H = \-j_ay + 1^6 + = Hg + H^ + H^ + . . . + H^ + . . . . where H k i s a homogeneous polynomial i n a,£,y and 6 of degree k. Let ay= § and p6 = Q. Now we w i l l prove that H^ i s a homogeneous polynomial of degree -g- i n £, i f 7 6 k i s even and = 0 i f k i s odd For Hg t h i s i s t r u e . L e t us make the i n d u c t i v e assumption that H = 0 I f l i s odd and *t = •  P 0 ? Z + P X ? ^ 1 C + • • • I *i? > 2. i f t i s Now l e t Ca P i_q r.s f3^Y & > where P + q + r + s = k ; > t he a ,. term of H k. From (3-22) we have ( 3 . 2 3 ' ) aH a = ( a H ^ + ... + a H ( k - l ) a } + a H k a + . . . . ( 3 - 2 3 " ) yH • Y = ( Y H 2 y + ... + Y H ( k - l ) Y } + Y H k y + . . . . (3.24') = (PH2r3 + • • • + P H(k-l)£j) + ^ H k P • • • • ( 3 . 2 4 " ) 6H 6 = ( 6 H26 +'••• + 5 H ( k - l ) 5 ) ^ + 6 I k 6 . + . . . . From the i n d u c t i v e assumption i t i s c l e a r that the expressions i n the parenthesis i n ( 3 . 2 3 ) and (3.24) are homogeneous polynomials i n § and Q. A f t e r s u b s t i t -u t i n g the r i g h t hand sides of ( 3 . 2 3 ) and (3.24) i n t o ( 3 . 2 1 ) . and making; user-of ( 3 - 1 0 ) , we w i l l equate the c o e f f i c i e n t of a?h^^FM'a ;wb,.ere.i;p+Hq+r+:s := -k -to;- zero.^.hlch i s C(a o + b q + c r + d :s) plus the c o e f f i c i e n t of the • v o o o h y r k terms of the form KrC , where I + h = -| i f k i s even. There w i l l be no-added c o n t r i b u t i o n i f k i s odd. The c o e f f i c i e n t of Z,1^, where i-+ h = ^  w i l l he taken i n t o c o n s i d e r a t i o n only i n the case when p = r and q = s i n which case our a s s e r t i o n i s t r u e . Now C ( a o p + b o q + V + d o s ^ = c ( ^ ( P - r ) - + \n(q-s)) which w i l l be zero only I f C = 0 or i f X^(p-r) + X m(q-s) = 0 The l a t t e r i s p o s s i b l e o n l y when p = r and q = s. Both cases, prove our a s s e r t i o n and consequently H has the desired form. Obviously, cxH^ =*-y&- and 0Hp = 6Hfi. Therefore ( 3 . 2 1 ) becomes 7 7 (3-25) a H Q(a + c) + pHp(b + d) = 0 i d e n t i c a l l y i n CX,|3,Y, and 6. This i m p l i e s t h a t a + c = 0 and b + d = 0 . I f i n a d d i t i o n we suppose that (3.26) a Q = lyQ and P q = ±EQ , we can prove that a and b are pur e l y imaginary. We r e c a l l the r e l a t i o n between the old v a r i a b l e z* and the new variable"' z i (3-27) z* = C- tz 1 + G 2 z 2 + ... + C 2 n z 2 n and the c o n d i t i o n s ( i ) I f X k = a + i b , ab =(= 0 , then T k = and C~k = C^, 1 _< k, i <. n or n + 1 _< k, i <_ ( i i ) I f \ k i s pur e l y imaginary, then \ = -X k = 1 < k < n C ^ = i C k o r C k = !C - t,£ = k + n I n the equation z* = JH „ z* the c o e f f i c i e n t s on the r i g h t hand side are r e a l and i f z* given by (3.27) i s a complex s o l u t i o n , then z* = + . . . + w i l l a l s o be a s o l u t i o n . I f i n case ( i ) we set (3.28) Z l « \ and i n case ( i i ) (3.29) 78 ( 3 . 3 0 ) 2 = 4- C2Z:2 + ... + C 2 n Z 2 n i s a l s o a s o l u t i o n of the o r i g i n a l system. From the d e f i n i t i o n of Zfc, we can express them as power s e r i e s i n C7,P/YJ"& V 1 Z : i Y + i a + i"6 + i P + 0 + i f k = 1 i f k = n+1 i f k = m i f k = n+m otherwise I f we l e t i Y = a*, i p = 6*, i a = Y* and i"6~ = &*, then I I ' r a* + Y* + p* + 6* + 0 + i f k = 1 i f k = n+1 i f k = m i f k = n+m otherwise The consequence of assumption I I I which we w i l l c a l l I I I ' i s : Z._ , Z, . Z T . and Z , do not contain any term / I 3 m3 n+13 n+m J of the form a* Pp* qY* r6* S,9 p = r+1 and q = s; q = s+1 and p = r; r = p+1 and q = s; s = q+1 and p = r r e s p e c t i v e l y . From ( 3 - 9 ) we see that i a = a i a i p = h i p i"y; .= "ci"Y i"5 = l i b or Y9. (3.31) •• '--¥•*.••=/ ay* a* = f . ca* , p* = dp* 6* = b6* ' Put c(a , p,Y>0 = a*(a * , P*,y*,6*) and a ( a , 6 , Y , 6 ) = c*(a*,p*, Y*,6*) w i t h s i m i l a r r e s u l t s f o r b and d. So (3-31) has the form a* = a*(a*,p*,Y*,6*)a* p* = b*(a*,p*,Y*>6*)p* Y* = G*.(O* , P * , Y*»6*)Y * 6* = •d*(ci*,"p*,Y*»:6*)6* Now conclusions I ' , I I ' , and:III' are the exact analogues f o r Z, a*>b*, c\*, and d* of the assumptions I , I I , and I I I f o r z, a,b,c, and d. Consequently, the unique c o e f f i c i - . ents f o r a , b , c , d and cpk^  determined by mathematical Induction remain unchanged f o r the analogues of a,b,c, d and cpk<t . Prom (3.26) i t f o l l o w s t h a t • Y „ = ibV • ; ~. ' ,r , ^ O . O and 6 C = i P Q . Hence a Q Y 0 = ^ a n d - - - p 6 8 0 = - 0 o 6 o , i . e . , a Q Y 0 and P Q 6 o are p u r e l y imaginary.,. Also ay = -ay = a*Y* and p6=-~66=p*8*. Thus a*Y* and 6*6* may be replaced by ay and 66 r e s p e c t i v e l y . We n o t i c e that a(a,p,y,6) = a(a*,6 * ,y*,6*) and, since a+c = 0 = b+d, we,have / a * ( a * , p*, y * i 6^)Vp:>:a;(a, pyyyS) • fc:-$C#i0>\Y, & ) = -a(a ,p,y,6) and b*(a*,p*, Y*,6*) = b(a , p, Y,6) = 3(a,p,Y,6) = -IS (a, p, Y , 6 ). bCv Thus a and h are p u r e l y imaginary. Now the equations at a = a Qe , Y = Y 0e -at p = p Q e b t , and 6 = 6 Q e " b t imply t h a t i y = i Y 0 e = aQe = a and i6 = p. But i y = a* e t c . , t h e r e f o r e a = a*, p = p*, y = y*> and 6 = 6*. F i n a l l y , p u t t i n g (3-33) Z k = z k l n (3.2b) and (3.29) we get p r e c i s e l y the r e a l i t y c o n d i t i o n s f o r z*. Hence (3.26) leads t o a r e a l s o l u t i o n z* of (3.7). 5.5 Convergence of the S o l u t i o n . Our convergence proof i s modelled on that of SIEGEL [ l ] . I In view of the assumption I I we may w r i t e the formal s o l u t i o n as ; f o r k = 1 f o r k = m = ( ^  + z k " t o r k = n + 1 f o r k = n+m otherwise where the z^ are poser s e r i e s i n O,,P,YJ and 6 beginning w i t h quadratic terms and a,p,y,6 s a t i s f y i n g (3.9). The power s e r i e s f o r a,b,c, and d can be w r i t t e n i n the form (3.34) z k * fa + z k P + z k *• Y + z k 6 + z k * 0 + z k (3.35) a b c d X± + a* V + b * -X]_ + c* -X m + d* -X 1 - a* -X - b* m^ 8 1 I n s e r t i n g .(3.3*0 i n t o (3.7) and us i n g (3-9) we o b t a i n *k = z k a d + Z k ^ + Z k Y Y + Z k 6 * - a ( a 2 k a " Y Z k Y ) "+" b ( P 2 k P " 6 Zk6> k =s 1,2,..., 2n. So we have (3.36) a ( a Z k a - Y z k Y ) . + b ( p z k p - 6z;k6.) - X k z k . f f c Now i f we s u b s t i t u t e (3.3*0 and (3-35) i n t o (3-36) and compare the c o e f f i c i e n t s of aP&^Y*&S> where p + q + r + s > l , on both side s of these equations, we o b t a i n (3.37-) X ^ c e z ^ - YZ| y - z j ) * X j p z J p - 6z* 6) + • f l " YZj y) - b * ( p z j p - 6z* 6) ("7"> (^az*, - Y z £ Y ) + X m O z ^ - 6 z * p - z j ) + b * P " fm " a * ( a z m a " • YZmY> " ^&zZfi ' ^ (3.37"') hla*U*.)* ~ yZW) j^f^^(n-Wr * z ? n + l ) • > " a * Y - W " a * ( a Z f n + l ) a " Y Z ? n + l ) Y } " ^ ^ 1 ) P " 6 z f n + l ) 6 ) ( 3 . 3 ^ ) H ( a z f h + m ) a " Y Z t n + m ) Y } + \n^ ztn +m)p " 6 z?n +m)6 + zfn -m)) " b * 6 - Vm - a*( a ztn +m)a " Y Z f n + m ) Y ) " b*^ Z?n+m)p " 6 zfn+m)6) 8 2 (3.37V) X ^ a z k - yz\y) + X m ( 0 z * p - 6z* 6) - x^z* . - f k ~ a * < a Z k a " Y Z k Y ) " b * ^ Z k 0 " 6 z k 6 ) where k =f= l,m,n+l and n+m I n the sequel we s h a l l use the f o l l o w i n g n o t a t i o n : I f h i s a power s e r i e s expression i n a,$,y and 6, then (h) w i l l denote the c o e f f i c i e n t of a p p q Y r 6 S . v ' p qr s We s h a l l a l s o r e f e r t o k=l,m,m+n or.n+1 as exc e p t i o n a l cases. From (3-37V) we ob t a i n (3.38) {X 1(p-r) + X m(q-s) - X k } ( z * ) p q r s = ( f k ) p q r s -min(p,qi-r , s ) . (p~r) A ( & ' l l l l ( Z * ] s ) 2*1,(1-1^-1,3-1 min(p,q,r , s ) (q-s) ^ ( h > i i l i ( z * ' p - i , q - l , r - t , s - l since CO a* m and we keep i n view (3.7)• I n the e x c e p t i o n a l cases X 1(p-r) + X m(q-s) - X k » 0 However i f | p - r | f 1 and q f s , then ( a * a ) p ( | r s = 0 * ( C * Y W s * a n d l f l ^ " 6 ! ^ 1 a n d r v> t n e n O ^ p q r s (d *6) = 0 . Hence (3.38) remains a p p l i c a b l e i n these > 'pqrs v ' cases. I f p = r+1 and q = s, then { X ^ a z ^ - y z J Y - z{) + X n ( P z J p - 5z* 5) } p q r g -= • by vi.-rtu"e.. of .a^Mmpittoni'I'Ii'.. Tlie.^cothiffK^iQ.ases ;can-: be handled i n a s i m i l a r manner. A c c o r d i n g l y we have (3.39') ( a * ) r q r q - ( f l ) ( r + l ) q r q 83 (5.39") - ( a * ) r q r < 1 - ( f n + l ) r q ( r + l ) q (3.40.) 0>*) P S P S - ( f m ) p ( B + l ) p s (3.*0") - 0 > ' ) P I P G = ( W p s p ( s + 1 ) Thus, i f we are not In the e x c e p t i o n a l cases we have from (3.38) C - * 1 ) ( ^ p q r s - x j p ^ t q - s ) - . ^ min(p,r,q,s) X 1(p-r)+X t a(q-s)-X k min(p,q,r,s) t s ~ q ) i h . ^ lllVZk)v-t,a-lsT^l,B-l ^ 1 ( p - r ) + X m ( q - s ) - X k 1 =N We can f i n d a constant C^ > -A |. 1 | < c } | r - p l J , c X 1(p-r)+X m(q-s) 1 ' j \ 1(p--ry+X m(q-sy-X k| 1 U 1 ( p - r ) + \ m ( q - s ) - X m | Consequently from (3.41) I t f o l l o w s t h a t / min(p,q,r,s) C M 2 ) | ( 2 \ ) p q r s ! < C ^ l ( f k ) p q r s l + & I ( a * ) ^ u l • I min(p,q,r,s) \ I n the e x c e p t i o n a l cases we have the r e l a t i o n s 84 (3.^3») | l ( a * ) r q r q l < ° 1 ' ( f l ' (r+1) qrql ( 3 -43 " ) | l ^ ) r q r q l < c l ' f f n + l ' r q ( r * l ) q l (3.44<) | i ^ ) p s p S l < ° 1 ' (fm'p(s+'l)ps[ ( 3 - 44 " ) ° 1 ' ^ (n+m) ^ psp(s+l) But f k are power s e r i e s i n z - ^ Z g , . . . , z 2 n c o n v e r S l n S - ^ o r | z | < 2R, where R > 0, £ = l,2r,. ..,2n. Hence, there e x i s t s a constant Cg > 0 7^  1 C 2 f o r a l l | z j < R (-t,k = 1,2, ...,2n) I t can he shown hy Catachy'-s m a j o r i z i n g method th a t (3.^5) f k - ^ =: £ » ' ( 1 — & (1- f 2 ) .. . (1- f2n) ' R R R Now i f A and B are p o s i t i v e constants then 1 • 1 (1-Az 1)(1-Bz 2) 1-Az 1-Bz 2 I n general i f A,,A_,...A~ are such t h a t A. > 0 f o r ° 1* 2 2n l i=l,2,...,2n then ( l - A l Z l ) . . . ( 1 - A 2 n z 2 n ) l - A ^ . . - A 2 n z 2 n Consequently / G ? C 2 X ~ R x ~ R 2n where z = . £, z. i = l i Since f^, begins w i t h quadratic terms, we can w r i t e 8 5 (3.46) f k < l — % - C„ - - 2 -K 1 - | 2 R z 2 \ = C,. fo'r some C,, Ch. > 0 Let us m u l t i p l y (3-42) by | a p | | 0 | A J Y | r | - 6 | S , (3.43') by |a r + 1 H P l q l Y | r | « | q , (3.43") by |a|r|p| q| Y| r + 1| 6 | \ (3.44«) by | a | P | p | S + 1 | Y r | 6 | S , (3-44«) by |a | P | 0| S | Y | P | 6 | 3 + 1 and sum the i n e q u a l i t i e s over p,q,r,s and k from 1 t o 2n (except f o r k = l , p=r+l and q=s; k=m,; p=r and q=s+l; k«n+l, r=p+l and q=s; k=n+m, p=r and q=s+l) . F i n a l l y add a l l the f i v e i n e q u a l i t i e s so obtained. Now replace i n the expression z| + z| + ... + z* n the c o e f f i c i e n t s -by t h e i r absolute values and a3 0, y and 6 by |a| = |B| = | Y | = | 6 | = -|s a n <^ denote the r e s u l t so obtained by § Z , where Z i s a power s e r i e s i n § wi t h p o s i t i v e c o e f f i c i e n t s Let A and B denote the expressions obtained from a* and b* r e s p e c t i v e l y by these s u b s t i t u t i o n s . Thus i n view of (3-46) we have 2nC ( § + § Z ) 2 § Z + § ( A • • + B ) < C n [(A + B ) ? Z + — ^ ] 1 - C 4 ( ? + 5Z) or ( 2 (3 . 4 7 ) Z + (A + B ) - C C 5 [ ( A + B)Z + | ( * ;z ) 1 where C r > 0 i s a s u i t a b l e constant 5 Put A + B + z = ~y~ Since a l l the c o e f f i c i e n t s are p o s i t i v e , it-"turns'out t h a t (A + B ) Z ~<1 y 2 and • Z Y" and consequently from (3-47) we have 86 (3.48) v ^ c 6 [ v 2 + s + ggv+ sv! 3 f o r a s u i t a b l e Cg > 0 . L e t t i n g (1 + g ) ~ y = X, we see that \^<*<^X where i - c 6 ( ? + c y ) Ve m u l t i p l y (3.48) by 1 + g i n order to get i - c6(g + ? y ) x ^ c - ^ ( 1 + + + g) + + g) ^ '6 - - - - - - r. or x < y x2 + 2 g x + g2 + g 6 i - c 6 ( g + gy) ^ c (g+ x)2 + g  ^ 6 i - c6(g + gy+y) Consequently g + X ^ g + C + (S + X ) 2  0 1 - c6(g + X ) Now i f we assume Cg > 1, then g + X ~ < 2 C g g + ^ + X ) 2 -i - c6(g + X ) or .2 W -<^ c -LJLJL-.^ where ¥ = 5 + X and C = 2C r • 1 - CW " ° or 2 V ^C2 + V where V = CW and C 2g = T 1 - V Suppose ¥ i s a power s e r i e s i n T without a constant term and with, p o s i t i v e c o e f f i c i e n t s , s a t i s f y i n g 87 2 (3.49) II = T + U 1 - IT Let. U converge f o r |T| < P. Then V a l s o converges f o r |T| < P.. Since V ^ U . Now (3.49) i s equivalent t o 2 2U - U + T - 0 which may he w r i t t e n as 16CT2 - 8U + 8T = 0 or as (1 4 U ) 2 1 - 8T which i m p l i e s that 1 + 8u + ...*<C— and th e r e f o r e 1 - 8T 1 - 8 T T 1 But the s e r i e s converges i f I Ti < -a . 1 - 8T l i - B Thus hy t r a c i n g hack the argument we can see tha t W as a power s e r i e s i n § w i l l converge f o r |§[ < — . F i n a l l y , we conclude t h a t the s e r i e s f o r z^, z^, . .., z 2 n , a and b w i l l converge f o r | a | , | £ | , | y U I S I < — ^ • l 6 C ^ I n view of the f a c t t h a t (3.50) a = a o e a t , P = 3 Q e " b t -at . t -ht Y - V , - 6 Qe a = + e t c . and a and b are p u r e l y imaginary, the sums f o r zl>'''z2n.} a a n d b converge a b s o l u t e l y i f | a Q | , | P Q ! J l y o ^ l 6 o '  < iSC^' I f > " t h e s e c o n d i t i o n s are s a t i s f i e d and we s u b s t i t u t e (3-50) i n t o (3.9) we get a F o u r i e r s e r i e s expansion of the p e r i o d i c s o l u t i o n . 88 Remark. This method of c o n s t r u c t i n g a p e r i o d i c s o l u t i o n can be extended t o a system having more than two p a i r s of p u r e l y imaginary eigenvalues but the expressions Involved are l i k e l y t o become unmanageable. The End. 89 BIBLIOGRAPHY AIZERMAN, M. A. [ 1 ] , On a problem concerning the s t a b i l i t y i n the l a r g e of dynamical systems. Uspehi Mat. Nauk (N. S . ) . 4 ( 2 8 ) , 1 8 7 -1 8 8 (1949) BARBASHIN, E. A . [ l ] , On the S t a b i l i t y of s o l u t i o n s of a non-l i n e a r equation of T h i r d order. P r i k l . Mat. Meh. 16, 6 2 9 - 6 3 2 ( 1 9 5 2 ) " and KRASOVSKII, N. N. [ l ] , On the s t a b i l i t y of the motion i n the l a r g e Dokl. Akad. Nauk. SSSR. No. 8 6 , 4 5 3 - 4 5 6 ( 1 9 5 2 ) . 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T.CXVTI, 184 -187 (1957) • 9 3 SIMANOV, S.-N. [1], S I E G E L , - C. L. [1], WINTNER, A. [ 1 ] , On S t a b i l i t y of Non-linear equations of the t h i r d order. P r i k l . Mat. Meh. IJ, 369 - 372. (1953) • Vorlesungen :Li.bT8^immelsmechanik, B e r l i n - Heidelberg and New York: ; S p r i n g e r - V e r l a g , 1956 pp. 69 - 92. The ..Analytic foundations of c e l e s t i a l Mechanics. P r i n c e t o n : "Princeton Univ. P r e s s , 1947. 

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