Existence and stability of periodic motion under higher order averaging (Q1100329)

Let \(L_{\epsilon}\) be a continuous and matrix valued function of \(\epsilon\). It is a very interesting question to study the effects on the asymptotic expansion of the eigenvalues of \(L_{\epsilon}\) when a perturbation of order \(\epsilon\) n is added. In particular it is said that \(L_{\epsilon}\) is k-hyperbolic if for every continuous \(N_{\epsilon}\) defined for \(\epsilon\geq 0\) satisfying \(N_{\epsilon}=0(\epsilon\) k), there exists an interval \(0<\epsilon <\epsilon_ 1\) in which \(L_{\epsilon}+N_{\epsilon}\) is hyperbolic of the same type. The author gives some sufficient conditions for that \(L_{\epsilon}\) be k-hyperbolic which generalize and extend some previous results. As an application he studies the stability of some periodic solutions for a coupled pair of Duffing equations.