Melnikov’s Criteria and Chaos Analysis in the Nonlinear Schrödinger Equation with Kerr Law Nonlinearity

Abstract

The dynamics of the nonlinear Schrödinger equation with Kerr law nonlinearity with two perturbation terms are investigated. By using Melnikov method, the threshold values of chaotic motion under periodic perturbation are given. Moreover we also study the effects of the parameters of system on dynamical behaviors by using numerical simulation. The numerical simulations, including bifurcation diagram of fixed points, chaos threshold diagram of system in three-dimensional space, maximum Lyapunov exponent, and phase portraits, are also plotted to illustrate theoretical analysis and to expose the complex dynamical behaviors. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting controller e, amplitude $d_1$, and frequency ${\omega }_2$ of external forcing which can be considered a control strategy, and when the frequenciesy ${\omega }_2$ and ${\omega }_1$ approach the maximum frequency of disturbance, the system turmoil intensifies and control intensity increases.