Linearization near the integral manifold for a system of differential equations with piecewise constant argument
Assuming exponential dichotomy conditions with respect to the continuous matrix function \( A \), the existence of a homeomorphism sending the solutions of the differential equation with piecewise constant arguments \[ x'(t)=B(t)x([t])+f(t,x(t),y(t)), \quad y'(t)=A(t)y(t)+C(t)y([t])+g(t,x(t),y(t))\tag{1} \] into solutions of the reduced equation \[ x'(t)=B(t)x([t])+f(t,x(t),v(t,x(t))), \qquad y'(t)=A(t)y(t)\tag{2} \] is proved. Here \(y=v(t,x)\) describes an integral manifold of (1) whose existence is shown in a former paper. It is assumed that the continuous matrix functions \( B \) and \(C \) are sufficiently small and that the continuous nonlinearities \(f \) and \(g \) are small Lipschitzian. The admissible bounds for \( B \), \(C \) and the admissible Lipschitz constants for \( f \) and \( g\) are expressed in terms of the bound for \( A \) and its dichotomy properties.
scientific article; zbMATH DE number 1115418