  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.
