On invariant manifolds of linear differential equations. II

TL;DR

This paper extends the theory of invariant manifolds for linear differential equations, introduces a notion of equivalence between equations of different orders, and applies these results to periodic coefficient systems and nonlinear equations.

Contribution

It develops a new method for splitting invariant manifolds, introduces a concept of equivalence for linear equations of different orders, and applies these to periodic and nonlinear systems.

Findings

Representation of fundamental solutions with real matrices

Conditions for equivalence of linear differential equations of different orders

Application to reduction of nonlinear equations with periodic linear parts

Abstract

This is the continuation of previous article. For subspaces $M^n(t)$ and $M^{n-m}(t)$ which are invariant manifolds of the differential equation under consideration we build a change of variables which splits this equation into a system of two independent equations. A notion of equivalence of linear differential equations of different orders is introduced. Necessary and sufficient conditions of this equivalence are given. These results are applied to the Flocke-Lyapunov theory for linear equations with periodic coefficients with a period T. In the case when monodromy matrix of the equation has negative eigenvalues, thus reduction in $R^m$ to an equation with constant coeficcients is possible only with doubling of reduction matrix period, we prove the possibility of splitting off in $R^m$ of equations with negative eigenvalues of monodromy matrix with the help of a real matrix without period doubling. For the fundamental matrix of solutions of an equation with periodic coefficients $X(t), X(t)=E$, we find representation $X(t)=\Phi(t)e^{Ht}\Phi^{+}(0)$ with real rectangular matrices $H$ and $\Phi(t), \Phi(t)=\Phi(t+T)$. We bring two applications of these results: 1) reduction of nonlinear differential equation in $R^n$ with distinguished linear part which is periodic with period T to the equation in $R^m, m>n$, with a constiant matrix of coefficients of the linear part; 2) for introdusing of amplitude-phase coordinates in the neigbourhood of periodic orbit of autonomous differential equation with separation of the linear part with constant matrix of coefficients.