Response solutions to the quasi-periodically forced systems with degenerate equilibrium: A simple proof of a result of W. Si and J. Si. and extensions

TL;DR

This paper presents a simplified, regularity-preserving fixed point approach to prove the existence of response solutions in quasi-periodically forced systems with degenerate fixed points, extending previous KAM-based results.

Contribution

The authors introduce a contraction mapping method that simplifies the proof, reduces regularity loss, and applies to finitely differentiable problems, also exploring higher-dimensional cases and new forcing scenarios.

Findings

Response solutions exist under the new method.

The approach maintains the same regularity as forcing.

New phenomena like monodromy are observed in complex cases.

Abstract

We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with a degenerate fixed points. The same questions were answered in \cite{ss18} using two versions of KAM theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a contraction. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forcing. Moreover, the method here applies when problems are only finitely differentiable. It also weakens slightly the non-degeneracy conditions. Since the method is based on the contraction mapping principle, we also obtain automatically smooth dependence on parameters and, when studying complex versions of the problem we discover the new phenomenon of monodromy. We also present results for higher dimensional systems, but for higher dimensional systems, the concept of degenerate fixed points is much more subtle than in one dimensional systems. To illustrate the power of the method, we also consider two problems not studied in \cite{ss18}: the forcing with zero average and second order oscillators. We show that in the zero average forcing case, the solutions are qualitatively different, but the second order oscillators is remarkably similar.