Existence of periodic solutions of differential systems with small parameters (Q949298)

The method of reflecting functions is used to study periodic solutions of periodic systems with small parameters. Given the differential equation \noindent \((1)\)\hfill \(x'=X(t,x),\quad t\in \mathbb{R},\,\,x\in D\subset \mathbb{R}^n\,,\) \hfill {} \noindent then the continuously differentiable vector function \(F(t,x)\) on \(\mathbb{R}\times \mathbb{R}^n\) is called a reflective function [\textit{V. I. Mironenko}, Differ. Equations 28, No.6, 789-794 (1992); translation from Differ. Uravn. 28, No.6, 984-991 (1992; Zbl 0834.34048)] if it is solution of the Cauchy problem \[ F_t(t,x)+F_x(t,x) X(t,x)+X(-t,F(t,x))=0,\quad F(0,x)=x\,. \] In this paper the reflective functions are applied to study the existence of periodic solutions of periodic systems with small parameters. The following result is a typical one (the paper establishes several results of this type): Theorem. Suppose that \(F(t,x)\) is the reflective function of the \(n\)-dimensional \(2\omega\)-periodic system \[ x'=X(t,x,0)\,, \] and there is a \(\bar x\) satisfying \(F(-\omega,\bar x)=\bar x\) and \(\text{ det\,}(F_x(-\omega,\bar x)-E)\neq 0\). Then, for \(| \varepsilon| \) sufficiently small, the periodic system \[ x'=X(t,x,\varepsilon)+F_x^{-1}(t,x) P(t,x)-P(-t,F(t,x)) \] has a \(2\omega\)-periodic solution.