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<p>In the current study, the nonlinear chains of atoms model will be solved using the improved modified Sardar sub-equation method. In this method partial differential equations can be transformed into nonlinear ordinary differential equations by using a certain wave transformation. The proposed method, will offer simple calculations, high accuracy, minimal processing effort, and a wide range of solution forms. Travelling wave solutions involving trigonometric, hyperbolic, and exponential functions will yield bright, dark, singular, M-shaped, W-shaped, bell-shaped, and anti-bell shaped soliton solutions. Bifurcation analysis, chaotic behavior and sensitivity analysis of the proposed model will be examined using planar dynamical system technique. The model exhibits periodic, quasi-periodic, and chaotic behaviors. Additionally, the linear stability procedure is used to evaluate the stability of the model. The proposed method will be an effective way to obtain exact solutions to a variety of nonlinear partial differential equations. A series of 3<i>D</i>, 2<i>D</i> and contour graphical representations will be employed.</p>

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Bifurcations, Chaotic Behavior, Sensitivity Analysis and Investigation of Analytical Soliton Solutions to the Nonlinear Chains of Atoms Model

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年5月15日

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  • Shahida Parveen,
  • Tahir Nazir,
  • Muhammad Abbas,
  • Asnake Birhanu,
  • Y. S. Hamed,
  • Muhammad Zain Yousaf

摘要

In the current study, the nonlinear chains of atoms model will be solved using the improved modified Sardar sub-equation method. In this method partial differential equations can be transformed into nonlinear ordinary differential equations by using a certain wave transformation. The proposed method, will offer simple calculations, high accuracy, minimal processing effort, and a wide range of solution forms. Travelling wave solutions involving trigonometric, hyperbolic, and exponential functions will yield bright, dark, singular, M-shaped, W-shaped, bell-shaped, and anti-bell shaped soliton solutions. Bifurcation analysis, chaotic behavior and sensitivity analysis of the proposed model will be examined using planar dynamical system technique. The model exhibits periodic, quasi-periodic, and chaotic behaviors. Additionally, the linear stability procedure is used to evaluate the stability of the model. The proposed method will be an effective way to obtain exact solutions to a variety of nonlinear partial differential equations. A series of 3D, 2D and contour graphical representations will be employed.