A KAM-Theorem for Persistence of Quasi-periodic Invariant Tori in Bifurcation Theory of Equilibrium Points

TL;DR

This paper proves a KAM-theorem applicable to differential equations with degeneracies, ensuring the persistence of quasi-periodic invariant tori during complex bifurcations in dynamical systems.

Contribution

It introduces a KAM-theorem for finitely differentiable systems with degeneracies, extending the understanding of invariant tori in bifurcation scenarios.

Findings

Proves persistence of quasi-periodic tori in multiple Hopf bifurcations.

Handles degeneracies in differential equations.

Applicable to equilibrium point bifurcations.

Abstract

In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in multiple Hopf and zero-multiple Hopf bifurcations, as well as their subordinate bifurcations, of equilibrium points of continuous dynamical systems.