The Green function and conditions for the existence of invariant sets of impulse systems (Q752267)

We consider the system \[ d\phi /dt=a(\phi),\quad dx/dt=A(\phi)x+f(\phi,x),\quad \phi \in {\mathcal T}_ m\setminus \Gamma,\quad \Delta x|_{\phi \in \Gamma}=B(\phi)x+g(\phi,x), \] where \(x\in {\mathbb{R}}^ n\), \(\phi\in {\mathcal T}_ m\), \({\mathcal T}_ m\) is an m-dimensional torus and \(\Gamma\) is a one-dimensional smooth submanifold of \({\mathcal T}_ m\). We study the problem of existence of piecewise smooth invariant toroidal manifolds of the system. For that we define the Green function of the linearized equation in a neighbourhood of the torus and investigate its analytical properties.