Melnikov analysis and chaos control of nonlinear dispersive KdV equation under external periodic perturbation

Abstract

The behavior of non-smooth solitary waves switching to chaos is studied. Firstly, we present some singular homoclinic orbits of an unperturbed system. These singular homoclinic orbits correspond to non-smooth solutions. Secondly, we find that the peculiar solitary waves are more likely to be chaos by using the Melnikov theory. Finally, chaos thresholds under different amplitudes and frequencies of a periodic perturbation are given. One interesting finding is that there exists a peculiar perturbation frequency, which has significant effect on the system. The system is not well-controlled under this frequency. However, the system can be well controlled, when the frequency of the perturbation surpasses the peculiar perturbation frequency with fixed parameters of the unperturbed system.

On the Symmetries and Conservation Laws of the Multidimensional Nonlinear Damped Wave Equations

“…In this context, Cajori and Farlow obtained the first exact solution to the linear wave equation [6,7]. Further researches have been conducted to investigate the nonlinear wave equations where different approximate and numeric methods have been developed; see, for example, [8][9][10][11][12][13][14][15].…”

Chaos Analysis and Control of Relative Rotation System with Mathieu-Duffing Oscillator

“…Wu et al [20] studied stochastic chaos and its control by the top Lyapunov exponent. Yin et al [21] proved the peculiar solitary waves are more likely to be chaos by using the Melnikov theory and found that the system can be well controlled when the frequency of the perturbation surpasses the peculiar perturbation frequency with fixed parameters of the unperturbed system. Noise, as random phase, has been used in studying the control of chaos in the paper.…”