Study of Fixed Points and Chaos in Wave Propagation for the Generalized Damped Forced Korteweg-de Vries Equation using Bifurcation Analysis

Abstract

In this article, we consider the Generalized Damped Forced KdV (GDFKdV) equation given by $Ut + PUnUx + QUxxx + SU = γF(U, x, t, vi)$, where n ∈ N ; P, Q, S denote non-linear, dispersion, damping coefficients, respectively. We consider a forcing term F(U) = U(U − v1)(U − v2), where v1 and v2 are free parameters. We investigate the behaviour of fixed points evaluated for the corresponding dynamical system of our model problem. With respect to these fixed points, we investigate the effects of few significant the parameters involved in the model, namely, the free parameters v1 and v2, the nonlinear, dispersion and damping coefficients denoted by P, Q and S respectively, using the bifurcation tools. We also obtain the wave plots for the critical values of the nonlinear and dispersion coefficients for which the system becomes unstable and exhibit chaotic behaviour. We confirm the chaos in our dynamical system under various conditions with the help of Lyapunov exponents.