On exponential dichotomy and invariant sets of impulsive systems (Q2751652)

The author considers an impulsive system of the form NEWLINE\[NEWLINE{d\varphi \over dt}= a(\varphi,x,\varepsilon), \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{dx\over dt}= A(\varphi, \varepsilon)x+ f(\varphi,x,\varepsilon),\;\varphi\in\mathbb{T}_m\setminus \Gamma,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x|_{\varphi \in\Gamma}= B(\varphi, \varepsilon)x+ g(\varphi,x, \varepsilon), \tag{3}NEWLINE\]NEWLINE with \(x\in\mathbb{R}^n\), \(\varphi\in \mathbb{T}_m\), \(\mathbb{T}_m\) is an \(m\)-dimensional torus, \(\Gamma\) is a smooth compact submanifold of \(\mathbb{T}_m\) of codimension 1, \(\varepsilon\in \mathbb{R}\) is a small parameter. For the linearized system NEWLINE\[NEWLINE{d\varphi \over dt}=a_0(\varphi),\tag{4}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{dx\over dt}= A_0(\varphi) x,\;\varphi\in \mathbb{T}_m\setminus \Gamma,\tag{5}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\Delta x |_{\varphi \in\Gamma}= B_0(\varphi)x \tag{6}NEWLINE\]NEWLINE an exponential dichotomy is defined. For small perturbation of (2), the stability of this dichotomy is proved. NEWLINENEWLINENEWLINEIf the linear system (4)--(6) has an exponential dichotomy, sufficient conditions for the existence of invariant sets to (1)--(3) are derived.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00041].