Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation

Zhao Li; Shan Zhao; Zhao Li; Shan Zhao

doi:10.3934/math.20241100

AIMS Mathematics

2024, Volume 9, Issue 8: 22590-22601. doi: 10.3934/math.20241100

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Research article

Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation

Zhao Li ^{1
,
,},
Shan Zhao ²

1.
College of Computer Science, Chengdu University, Chengdu 610106, China
2.
School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu 610106, China

Received: 17 June 2024 Revised: 13 July 2024 Accepted: 19 July 2024 Published: 22 July 2024
MSC : 34H10, 35B20, 35C05, 35C07

In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented.
- Akbota equation,
- solitary wave solution,
- bifurcation,
- Lyapunov exponent,
- Chaos behavior,
- sensitivity analysis,
- traveling wave solution
Citation: Zhao Li, Shan Zhao. Bifurcation, chaotic behavior and solitary wave solutions for the Akbota equation[J]. AIMS Mathematics, 2024, 9(8): 22590-22601. doi: 10.3934/math.20241100

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Abstract

In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented.

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