Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative
- 1College of Computer Science, Chengdu University, Chengdu 610106, China
- 2Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
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.
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BibTeX
@article{li2026bifurcation,
title = {{Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative}},
author = {Zhao Li and Ejaz Hussain},
journal = {Fractal and Fractional},
year = {2026},
month = may,
doi = {10.3390/fractalfract10050335},
url = {https://doi.org/10.3390/fractalfract10050335},
note = {CC BY},
}
RIS
TY - JOUR
TI - Bifurcation Analysis and Chaotic Behaviors of and a Traveling-Wave Solution to the Zhiber–Shabat Equation with a Truncated M-Fractional Derivative
AU - Zhao Li
AU - Ejaz Hussain
PY - 2026
DA - 2026/05/15
JO - Fractal and Fractional
DO - 10.3390/fractalfract10050335
UR - https://doi.org/10.3390/fractalfract10050335
AB - 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.
SN - 2504-3110
ER -