Nonzero Periodic Solutions of Nonlinear Systems of Ordinary Differential Equations with a Singular Matrix Multiplying the Derivatives

Abstract

where x ∈ En, S and A are constant matrices, S is in general a singular matrix, f(t, x, λ) is a 2π-periodic function of t, λ ∈ Em, λ is a parameter, t ∈ (−∞,∞), and Ep is the p-dimensional vector space. Further, we assume that f(t, x, λ) = C(t, x, λ) + D(t, x, λ), where C(t, x, λ) is a form of order s > 1 in the variables x and λ, D(t, x, λ) is a finite sum of forms of order > s in the same variables, C(t, 0, λ) ≡ 0, and D(t, 0, λ) ≡ 0. Consequently, x = 0 is a solution of system (1) for any λ. The problem on the existence of a periodic solution of system (1) for S = E (E is the identity matrix) was considered in [1–5]; an important requirement in this connection was that the generating solution is a nonzero solution. If the zero solution of system (1) is chosen as the generating solution, then the zero solution is the unique periodic solution whose existence was guaranteed in the above-mentioned papers. The same problem was considered in [6]. The existence and uniqueness theorem for a periodic solution was proved there by the method of successive approximations without using any properties of the principal solution matrix of the linear approximation system. The contraction mapping principle was used in [7] to obtain conditions for the existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial with a prescribed number of harmonics and a remainder term. The existence of a periodic solution of system (1) in the form of a sum of a trigonometric polynomial and a series remainder was proved in [8, p. 375; 9, p. 202 of the Russian translation] by the method of successive approximations; the coefficients of the polynomial were found from an equation that is transcendental in general. Note that the papers [6–9] do not answer the question as to whether the periodic solution whose existence was proved in these papers for the case in which x = 0 is a solution of system (1) for any value of the parameter λ is nonzero. The problem on the existence of solutions and periodic solutions of system (1) was considered in [10–14] in the linear case under the assumption that S can be a singular matrix. In the present paper, we pose the problem of finding conditions for the existence of a nonzero periodic solution of system (1) in a neighborhood of the zero solution. The existence of a periodic solution in the form of a trigonometric series is proved by a fixed point method under a necessary condition for the existence of a solution of some finite-dimensional algebraic system of equations. We introduce the following notation: |u| = maxi {|ui|}, ‖Q‖ = max|u|≤1 |Qu|, u ∈ Ep, Q is a matrix, N is the set of positive integers, and N0 is the set of nonnegative integers. Let Mn be the set of series a0 + ∑∞ k=1 ak cos kt + bk sin kt, where a0 and ak, bk (the coefficients of the series) are n-dimensional vectors for any k ∈ N . The zero element of the set Mn is defined as the series with zero coefficients and is denoted by 0. Let x = a0 + ∑∞ k=1 ak cos kt + bk sin kt ∈ Mn and y = c0 + ∑∞ k=1 ck cos kt + dk sin kt ∈ Mn, let Q be an n× n matrix, and let γ be a real number. Then the operations x+ y, Qx, γx, and ẋ are defined by the formulas