Bifurcation Analysis of Time-Delayed Jerk Model

Articles in Press

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.

2 Department of Computer Science, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.

Abstract

This paper presents a comprehensive investigation into the dynamic behavior of a time-
delayed jerk model. The study introduces an innovative approach to delayed feedback
control, thoroughly examining the effects of delay on the system’s dynamics. The findings
reveal that the presence of delay can lead to the emergence of previously unrecognized
dynamic phenomena, such as Hopf, Bautin, and double-Hopf bifurcations. By employ-
ing the normal form method, the coefficients for the normal forms of each bifurcation
are determined, highlighting that the inclusion of delay significantly increases the sys-
tem’s complexity. Numerical simulations are conducted to validate the effectiveness of the
proposed delayed feedback control system, demonstrating its high accuracy in managing
complex and nonlinear dynamics. This study offers an in-depth analysis of the system’s
dynamic behavior while considering two distinct parameters, including co-dimension 1,
co-dimension 2 analyses, and the basin of attraction. Poincaré sections and Lyapunov ex-
ponents serve as essential tools for exploring the system’s dynamic behavior. The findings
of this research can assist designers and engineers in effectively addressing delay effects
in the design of mechanical and electrical systems, thereby enhancing the performance of
dynamic systems.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 12 October 2025

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History

  • Receive Date: 21 March 2025
  • Revise Date: 19 August 2025
  • Accept Date: 06 October 2025

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