Order to chaos transitions in damped KdV equation modeled as a jerk equation

The possibility of finding chaos in a KdV like system in the absence of any external forcing is explored without reducing its order by treating it as a third order Jerk equation. While bounded solutions with periodic nature exist for both right and left traveling wave solutions of the KdV equation, inclusion of terms that signify damping lead to chaotic behavior only for left moving waves. The bifurcation diagram obtained for suitable choice of the parameters shows interesting phenomena such as Hopf and period doubling bifurcations. To characterize the chaotic behavior, the spectrum of Lyapunov exponent is studied. Domains of stable and unstable solutions with the boundaries marking transitions have been identified in the parameter space.