Asymptotic stability of forced oscillations emanating from a limit cycle

TL;DR

This paper demonstrates that for analytic systems, the topological index of the bifurcation function determines the asymptotic stability of forced oscillations from a limit cycle, even when the derivative of the bifurcation function is zero.

Contribution

It generalizes classical stability conditions by showing the topological index can determine stability without relying on the derivative of the bifurcation function.

Findings

Topological index determines stability when derivative is zero.

Classical derivative-based conditions are complemented by topological methods.

Results apply to analytic systems near bifurcation points.

Abstract

Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the derivative of the corresponding bifurcation function F at the bifurcation point t. We show that for analytic systems this result is the one of topological nature, namely the topological index ind(t,F) is conclusive versus the sign of the derivative (d/dt)F(t). This allows to detect asymptotic stability also in the case when (d/dt)F(t)=0.