The dynamical playground of a higher-order cubic Ginzburg-Landau equation: from orbital connections and limit cycles to invariant tori and the onset of chaos

TL;DR

This paper explores the complex dynamics of a higher-order cubic Ginzburg-Landau equation, revealing how higher-order effects like third-order dispersion lead to diverse behaviors including chaos, with implications for understanding nonlinear wave phenomena.

Contribution

It demonstrates that third-order dispersion alone can induce complex spatiotemporal dynamics, including chaos, in the higher-order cubic Ginzburg-Landau equation, highlighting its dominant role.

Findings

Third-order dispersion can cause chaos without other higher-order effects.

Rich dynamics include equilibria, periodic, quasi-periodic, and chaotic solutions.

Phase space diagnostics effectively identify parametric regimes.

Abstract

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincar\'{e}-Bendixson--type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable low-dimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.