Variables Scaling to Solve a Singular Bifurcation Problem with Applications to Periodically Perturbed Autonomous Systems

TL;DR

This paper introduces a variable scaling method to analyze singular bifurcation problems in autonomous systems, enabling the application of the implicit function theorem without dimension reduction, and establishes existence and stability of periodic solutions.

Contribution

The novel approach uses variable scaling to handle singular bifurcations, avoiding traditional dimension reduction techniques like Lyapunov-Schmidt, and applies to periodic solutions in perturbed autonomous systems.

Findings

Existence of a unique solution branch established

Asymptotic stability of periodic solutions proven

Method applicable without dimension reduction

Abstract

By means of a linear scaling of the variables we convert a singular bifurcation equation in $\R^n$ into an equivalent equation to which the classical implicit function theorem can be directly applied. This allows to deduce the existence of a unique branch of solutions as well as a relevant property of the spectrum of the derivative of the singular bifurcation equation along the branch. We use these results to show the existence, uniqueness and the asymptotic stability of periodic solutions of a $T$-periodically perturbed autonomous system bifurcating from a $T$-periodic limit cycle of the autonomous unperturbed system. This problem is classical, but the novelty of the method proposed is that it allows us to solve the problem without any reduction of the dimension of the state space as it is usually done in the literature by means of the Lyapunov-Schmidt method.