Bifurcation study and pattern formation analysis of a nonlinear dynamical system for chaotic behavior in traveling wave solution

Jhangeer, Adil; Almusawa, Hassan; Hussain, Zamir

doi:10.1016/j.rinp.2022.105492

Cited by 31 publications

(13 citation statements)

References 51 publications

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“…In this section, we operate the Runge-Kutta method to scrutinize the sensitivity of the dynamical system described by PDS (85) [29]. The PDS (85) is chosen for investigation in this subsection utilizing sensitivity inspections.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…This subsection handles the multistability of a system like (88) for the perturbed term. To explore a dynamical system's tendency toward multistability (88), multistability [29][30][31] is the concurrent combination of numerous, even massive, solutions for a given range of physical variables and variable beginning values. For Remark 2 The study provided in this paper is completely new and has not been published before, as far as we are aware.…”

Section: Multistability Analysismentioning

confidence: 99%

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Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg-de Vries system of equations

Kopçasız

2024

Preprint

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This investigation discusses the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system of equations. The cmKdV equation describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore , after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to understand the dynamic behavior of these solutions better. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). The Maple software application is used to verify the correctness of all obtained solutions.

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Section: Sensitivity Analysismentioning

confidence: 99%

Section: Multistability Analysismentioning

confidence: 99%

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg-de Vries system of equations

Kopçasız

2024

Preprint

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“…Here, we adapt Equation ( 18) within the planer dynamical system by considering the Galilean transformation [53][54][55][56],…”

Section: Chaotic Analysis Of Lwementioning

confidence: 99%

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

Miah,

Alsharif,

Iqbal

et al. 2024

Mathematics

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In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis.

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“…[44] investigated the dynamical behavior of a time-space fractional Φ − 4 equation using the bifurcation approach of a planar dynamical system. In [45], bifurcation theory is used to study the phase plane analysis Chen-Lee-Liu equation with beta fractional derivatives. Wide-ranging investigations in chaotic analysis have been conducted over the last three decades [46,47].…”

Section: Introductionmentioning

confidence: 99%

Dynamical features and sensitivity visualization of thin-film Polarisation equation

Fatima,

Abbas,

Muhammad

2023

Phys. Scr.

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The present investigation describes the dynamical behavior, multi-stability, and traveling wave solutions of thin-film polarisation equations (TFPE) which describes the propagation of waves in thin-film ferroelectric materials. The extended direct algebraic technique is used to construct the traveling wave patterns. Visual representations of a few randomly selected solutions are provided for physical comprehension. The ordinary differential equation can be expressed in the planar dynamical system using the Galilean transformation. Using various initial conditions for the unperturbed dynamical system, phase portraits with various sorts of trajectories are created. Additionally, the Runge-Kutta method is used to plot nonlinear periodic waves and super nonlinear waves. Additionally, the Hamiltonian function for this undisturbed dynamical system is computed and shown. It also included the source term with amplitude and frequency parameters for the chaotic and quasi-periodic behaviors, and the system is also stated in the non-autonomous form. For the dynamical system under investigation, multi-stability is also thoroughly described. Furthermore, a full inspection of the sensitivity of the perturbed dynamical structure under various initial conditions has been conducted.

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Bifurcation study and pattern formation analysis of a nonlinear dynamical system for chaotic behavior in traveling wave solution

Cited by 31 publications

References 51 publications

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg-de Vries system of equations

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg-de Vries system of equations

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

Dynamical features and sensitivity visualization of thin-film Polarisation equation

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