Invariant manifolds of the Bonhoeffer-van der Pol oscillator

TL;DR

This paper investigates the stable and unstable manifolds of a saddle fixed point in the Bonhoeffer-van der Pol oscillator, linking homoclinic tangencies to chaos and bifurcation structures.

Contribution

It provides a numerical analysis of invariant manifolds and establishes a connection between homoclinic tangencies and chaotic behavior in the oscillator.

Findings

Homoclinic tangencies are associated with chaos and Smale horseshoes.

Non-chaotic zones lack homoclinic tangencies, regardless of Smale horseshoes presence.

Numerical study clarifies the bifurcation structure of the oscillator.

Abstract

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.