Bifurcation, chaotic pattern and traveling wave solutions for the fractional Bogoyavlenskii equation with multiplicative noise

Abstract

This paper presents a new study that incorporates the Stratonovich integral and conformal fractional derivative into the fractional stochastic Bogoyavlenskii equation with multiplicative noise. The study exposes the behavior of the system, including sensitivity, chaos and traveling wave solutions, by using the planar dynamical systems approach. Time series, periodic perturbation, phase portraits, and the Poincaré section are used to comprehensively study the dynamic properties. Notably, the research uses the planar dynamic systems method to build multiple traveling wave solutions, including kink wave, dark soliton, and double periodic solutions. Furthermore, a comparative study approach is applied to investigate the effects of fractional derivative and multiplicative noise on the traveling wave solutions, which demonstrate a significant influence of both variables. This work demonstrates the creative application of the planar dynamic system method to the analysis of nonlinear wave equations, offering insightful information that may be generalized to more complex wave phenomena.