Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

Abstract

The nonlinear dynamics of a generalized higher-order nonlinear Schrödinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bellshaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

New Exact Solutions, Dynamical and Chaotic Behaviors for the Fourth-Order Nonlinear Generalized Boussinesq Water Wave Equation

“…Then, we obtain det MðP 1 Þ = −β and det MðP 2 Þ = β. By the theory of planar dynamical systems [24][25][26][27][28][29][30], we draw the following conclusion as in Table 1. The different phase portraits of system (50) are shown in Figures 1 and 2.…”

“…The analysis of bifurcation and chaos behavior is a very interesting nonlinear phenomenon, which has been applied in many fields, such as engineering, telecommunication, and ecology [24][25][26][27]. By analyzing the dynamic behavior of differential equation, we can study whether the periodic external perturbation will lead to the chaotic behavior of differential equation.…”