Conditions for the instability of an invariant toroidal manifold of a discrete dynamical system in a Banach space (Q2754818)

The author studies a dynamical system \(x(n+1)=X(x(n))\), \(n\in\mathbb Z_{+}\), generated by the mapping \(X\in C^r(E \to E)\), where \(E\) is a Banach space. It is assumed that this system has a toroidal invariant manifold \(M=\{x\in E:x=f(\varphi),\;\varphi \in T^m:=\mathbb R^{m}/2\pi\mathbb Z^{m}\}\). Here \(f:T^m \to E\) is a \(C^r\) imbedding. Let \(E_\varphi \) denote the fiber at the point \(f(\varphi)\) of such a vector bundle over \(M\) that \(TE_{f(\varphi)}=TM_{f(\varphi)}\oplus E_\varphi \). It is assumed that the corresponding projector \(P(\varphi):E \to E_{\varphi }\) depends \(C^{r-1}\)-smoothly on \(\varphi \). Generalizing a result from \textit{A. M. Samoilenko, V. E. Slyusarchuk} and \textit{V. V. Slyusarchuk} [Ukr. Math. J. 49, 1872-1890 (1997; Zbl 0937.39005)], the author shows that the system under consideration is equivalent to the following one: NEWLINE\[NEWLINE\varphi(n+1)=\Theta \varphi(n),\quad \delta(n+1)=C(\varphi(n),\delta(n))\delta(n).NEWLINE\]NEWLINE Here \(\Theta :=f^{-1}\circ X\circ f\) and for any \(\varphi \in T^m\) and \(\delta \in E_{\varphi}\), \(\|\delta \|\leq\epsilon \) where \(\epsilon \) is sufficiently small, the mapping \(C(\varphi,\delta):E_{\varphi } \to E_{\Theta \varphi }\) is linear, bounded and \(C^r\)-smoothly depends on \(\varphi,\delta \).NEWLINENEWLINENEWLINELet \(C_1(\varphi):=P(\Theta \varphi)(dX)_{f(\varphi)}P(\varphi)\), \(C_{n}(\varphi):= P(\Theta^n\varphi)(dX)_{f(\Theta^n\varphi)}P(\Theta ^{n-1}\varphi)C_{n-1}(\varphi)\), \(n=2,\ldots\) Under the condition that \(C(\varphi,\delta)\equiv C(\varphi,0)\) the authors prove that the invariant manifold \(M\) is unstable, provided that NEWLINE\[NEWLINE\limsup_{n\to \infty }\sup_{\varphi \in T^m}\sup_{x\in E_{\varphi },\|x\|=1}\|C_n(\varphi)x\|=\inftyNEWLINE\]NEWLINE holds true.