Integral separation, Lyapunov transformations, and the small change in the direction of the solutions of periodic linear systems of ordinary differential equations (Q1364770)

A linear system \(dx/dt= A(t)x\) (where \(A(t)\) is a real-valued \(n\times n\) matrix) is called a system with integral separation if there exists a fundamental system of solutions \(x_1(t),\dots,x_n(t)\) such that \[ {|x_{k+1}(t)|\over|x_{k+1}(s)|}: {|x_k(t)|\over|x_k(s)|}\geq K e^{\alpha(t-s)} (k=1,\dots,n-1;\;K>0;\;\alpha>0). \] This is the case if and only if a mere small change in the direction of solutions may occur as a result of small perturbations of \(A(t)\). The system \(dx/dt= A(t)x\) is called Lyapunov reducible if there exists a transformation \(x= L(t)y\) with both \(L(t)\) and \(L(t)^{-1}\) uniformly bounded into a system \(dy/dt= By\) with constant matrix \(B\). The authors discuss these concepts for the particular case of periodical systems (the periods need not be fixed). In particular, the small change in the directions takes place if the characteristic multipliers are real and distinct or, in more generality, when they are distinct and their arguments do not change.