Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative
Year: 2026
Type: article
Abstract: In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, the bifurcation behavior of the equilibrium point of a two-dimensional dynamical system is discussed. At the same time, the dynamical behavior of a two-dimensional dynamical system with periodic disturbances is considered, revealing the complex chaotic phenomena of the system under specific parameters. A planar phase diagram, a three-dimensional phase diagram, a sensitivity analysis, and a maximum Lyapunov exponent diagram of the perturbed two-dimensional dynamical system were employed. Furthermore, various forms of accurate analytical solutions were obtained through traveling-wave transformation and numerical simulation. The three-dimensional, two-dimensional, density, and polar coordinates of the solutions were plotted using mathematical software. The results indicate that the fractional order and system parameters have a significant impact on the morphology and chaotic characteristics of the solution. This study provides new theoretical insights into the nonlinear dynamics of fractional-order Zhiber–Shabat equations.
Source: Fractal and Fractional
Authors: Zhao Li, Ejaz Hussain
Institutions: Chengdu University, University of the Punjab
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,258
OpenAlex ID: W7161293329