SADDLE-NODE BIFURCATION AND ITS CONTROL OF BURGERS–KdV EQUATION

Abstract

The Burgers–Korteweg-de Vries (KdV) equation had been used as nonlinear modes for acoustic shock waves in dusty plasmas and so on. The variable transformation and the Jacobi elliptic function method was introduced to find the exact solution. In this paper, we will research into the saddle-node bifurcation and its control of the forced Burgers–KdV. By the transformation, PDEs are reduced to ODEs. Analyzing the frequency response function and its unstable region of the trivial steady state, we know that the saddle-node bifurcation which leads to jump and hysteresis may appear in the resonance response. Controllers for bifurcation modification purpose are designed in order to remove or delay the occurrence of jump and hysteresis phenomena. By means of numerical simulations we compare the uncontrolled system with the controlled system and clarify that controllers are adequate for the saddle-node bifurcation control of the forced Burgers–KdV equation.