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UBC Theses and Dissertations

Contribution to nonlinear differential equations Lalli, Bikkar Singh


The subject matter of this thesis consists of a qualitative study of the stability and asymptotic stability of the zero solution of certain types of nonlinear differential equations, for arbitrary initial perturbations, and the construction of a periodic solution for a Hamiltonian system with n( ≥ 2) degrees of freedom. The material is divided into three chapters. The stability of the system (1) ẋ = xh₁(y) + ay, ẏ = f(x) + yh₂(x) with some restrictions on the functions h₁ (y), h₂(x) and f(x), is discussed in the first chapter. It turns out that some of the results proved by I.H. MUFTI ([l], [2], [3]), for the systems (2) ẋ = xh₁(y) + ay, ẏ = xh₂(x) + by and (3) ẋ = xh₁(y) + ay, ẏ = bx + yh₂(x) become particular cases of our results for system (1). Consequently an answer in the affirmative has been given to a problem proposed by I.H. MUFTI [1]. In the same chapter a generalization to the problem of M. A. AIZERMAN [l] for the case n = 2 is given in the form (4) ẋ = f₁(x) + f₂(y), ẏ = ax + f₃(y). This system has been discussed first by a qualitative method and second by constructing a LYAPUNOV function. In chapter II, stability of a quasilinear equation (5) [formula omitted] is discussed, by using LYAPUNOV's second method. It has been proved that if (i) [formula omitted] (ii) [formula omitted] for all values of x and y = ẋ (iii) [formula omitted] for all x,y (iv) [formula omitted] (where G,g and w are defined in Theorem 2.1) (v) [formula omitted] then the zero solution of (5) is asymptotically stable for arbitrary initial perturbations. In the same chapter certain equations of third order have also been discussed for "complete stability". These equations are special cases of (5) and are more general than those considered by SHIMANOV [l] and BARBASHIN [l]. AIZERMAN's [l] problem for the case n = 3 is generalized to two different forms, one of which is (6) [formula omitted] which is more general than the forms considered by V.A. PLISS [4] and N.N. KRASOVSKII [l]. Under a non-singular linear transformation equations(6) assume the form (7) [formula omitted] It has been proved that if in addition to the usual existence and uniqueness requirements, the conditions (i) [formula omitted] (ii) [formula omitted] (iii) [formula omitted] are fulfilled, then the zero solution of (7) is asymptotically stable in the large. In the third chapter a Hamiltonian system with n (≥ 2) degrees of freedom is considered in the normalized form (8)[formula omitted] where fĸ are power series in zk beginning with quadratic terms. A periodic solution for system (8) is constructed in the form (9) [formula omitted] where [formula omitted] is a homogeneous polynomial of degree [formula omitted] in terms of four time dependent variables a, B, y, õ. C. L. SIEGEL [l] constructs a periodic solution in terms of two variables [formula omitted] under the assumption that the corresponding linear system has a pair of purely imaginary eigenvalues. Here it is assumed that the linear system possesses two distinct pairs of purely imaginary eigenvalues and this necessitates the consideration of four time dependent variables in the construction of the periodic solution.

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