# Dissipative periodic and chaotic patterns to the KdV--Burgers and Gardner equations

## TL;DR

This paper analyzes the dynamics of the KdV-Burgers and Gardner equations with dissipation and external perturbations, revealing bifurcations, stability changes, and chaotic behavior through dynamical systems and Shil'nikov's analysis.

## Contribution

It introduces a reduced two-parameter dynamical system to study bifurcations and chaos in the KdV-Burgers and Gardner equations, including the application of Shil'nikov's theorem.

## Key findings

- Hopf bifurcation leads to stable limit cycles
- Homoclinic chaos observed in KdV-Burgers case
- Stable periodic orbits in Gardner equation

## Abstract

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.