Chaos in the Takens-Bogdanov bifurcation with O(2) symmetry

TL;DR

This paper investigates chaos near the Takens-Bogdanov bifurcation with O(2) symmetry, revealing cascades of period-doubling and Shil'nikov chaos through numerical and analytical methods.

Contribution

It demonstrates the presence of chaotic dynamics near the global bifurcation in the Takens-Bogdanov bifurcation with O(2) symmetry, extending previous analyses.

Findings

Chaos occurs arbitrarily close to the bifurcation point.

Cascades of period-doubling bifurcations are observed.

Shil'nikov-type chaos is identified near the global bifurcation.

Abstract

The Takens-Bogdanov bifurcation is a codimension two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens-Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and traveling and standing waves in the nonlinear regime and predicts the presence of modulated traveling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated traveling waves, the normal form for the Takens-Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension two point.