Dissipative Periodic and Chaotic Patterns to the Kdv--Burgers and Gardner Equations

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems. 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. In the case of the KdV-Burgers, we find homoclinic chaos by using Shil’nikov’s theorem