On local perturbations of linear extensions of dynamical systems on a torus (Q2722231)

The authors consider the following system on the direct product of \(m\)-dimensional torus \(T^m=\mathbb R^m/2\pi \mathbb Z^{n}\) and the space \(\mathbb R^{n}\): NEWLINE\[NEWLINE\dot\varphi =a(\varphi),\quad \dot x=A(\varphi)x.\tag{1}NEWLINE\]NEWLINE Here \(a(\varphi)\in C(T^m \mapsto \mathbb R^n)\), \(A(\varphi)\in C(T^m \mapsto \mathbb R^{n\times n})\). The system (1) is called the linear extension of the dynamical system \(\dot \varphi =a(\varphi)\) on the torus \(T^m\). The Green-Samojlenko function allows one to represent an invariant torus \(x=u(\varphi)\) of the nonhomogeneous system \(\dot\varphi =a(\varphi)\), \(\dot x=A(\varphi)x+f(\varphi),\) where \(f(\varphi)\in C(T^m \mapsto \mathbb R^n)\) in an integral form [see e.g. \textit{Yu. A. Mitropol'skij, A. M. Samojlenko} and \textit{V. L. Kulik}, Studies in dichotomy of linear systems of differential equations by means of Lyapunov functions. Kiev: Naukova Dumka (1990; Zbl 0776.34041)]. System (1) is called regular if it has a unique Green-Samojlenko function. The authors study the following problem: describe the class of matrix \(A(\varphi)\) perturbations (not necessarily small) which preserve the regularity of system (1).NEWLINENEWLINENEWLINELet \(\{\varphi_t(\varphi)\}\) be the flow on \(T^m\) generated by the first subsystem in (1). Denote by \(X(t;\varphi)\) a fundamental matrix of the system \(\dot x=A(\varphi_t(\varphi))x\) and suppose that there exist a matrix \(C(\varphi)\in C^1(T^m \mapsto \mathbb R^{n\times n})\), \(C^2(\varphi)=C(\varphi)\), and positive numbers \(K\), \(\gamma \) such that NEWLINE\[NEWLINE\|X(t;\varphi)C(\varphi)\|+ \|X(-t;\varphi)[C(\varphi)-\text{Id}]\|\leq Ke^{-\gamma t},\quad t\geq 0.NEWLINE\]NEWLINE The perturbed system in which \(A(\varphi)\) is replaced by \(A(\varphi)+B(\varphi)-C(\varphi)B(\varphi)(\text{Id}-C(\varphi))\) is regular provided that \(B(\varphi)\in C(T^m \mapsto \mathbb R^{n\times n})\) is an arbitrary matrix satisfying \(\int_{-\infty }^\infty\|B(\varphi_t(\varphi))\|dt<\infty\).