The existence of quasi-periodic invariant tori and double Hopf bifurcation of van der Pol's oscillator with delayed feedback

TL;DR

This paper investigates the bifurcation phenomena and the existence of quasi-periodic invariant tori in a delayed van der Pol oscillator, using advanced mathematical methods to analyze stability and bifurcation structures.

Contribution

It derives the normal form near the bifurcation point and proves the existence of invariant tori using KAM theory, providing new insights into delayed oscillator dynamics.

Findings

Existence of quasi-periodic 2-tori and 3-tori near bifurcation points.

Most parameter sets near the bifurcation exhibit these invariant tori.

Higher-order terms influence the stability and persistence of invariant tori.

Abstract

The double Hopf bifurcation and the existence of quasi-periodic invariant tori in a delayed van der Pol's oscillator are considered by regarding the damped coefficient and the delay as bifurcation parameters. Applying the center manifold reduction and the normal form method, we derive the normal form near the critical point and analyse the existence of invariant 2-tori and 3-tori for the truncated normal form. Furthermore, the effect of higher-order terms on these invariant tori are investigated by a KAM theorem, and it is proved that in a sufficiently small neighborhood of the bifurcation point, the delayed van der Pol's oscillator has quasi-periodic invariant 2-tori and 3-tori for most of the parameter set where its truncated normal form possesses quasi-periodic invariant 2-tori and 3-tori, respectively.