Existence and stability of limit cycles in a perturbed seventh-order differential equation

Meriem Ladjimi; Meriem Ladjimi

doi:10.3934/math.20251102

AIMS Mathematics

2025, Volume 10, Issue 10: 24901-24922. doi: 10.3934/math.20251102

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Existence and stability of limit cycles in a perturbed seventh-order differential equation

Meriem Ladjimi ^,

Department of Mathematics, Laboratory of LANOS, Faculty of Sciences, Badji Mokhtar–Annaba University, El Hadjar, 23000 Annaba, Algeria

Received: 12 August 2025 Revised: 16 October 2025 Accepted: 23 October 2025 Published: 30 October 2025
MSC : 34C25, 34C29, 37C15

This paper investigates the existence and stability of limit cycles in a class of perturbed seventh-order non-autonomous differential equations that model multi-frequency oscillatory behavior with damping, commonly encountered in mechanical and control systems. The high order of the equation presents significant challenges in detecting periodic solutions and analyzing their stability. By applying first-order averaging theory, we reduced the original equation to a lower-dimensional averaged equation. We then derived explicit conditions under which isolated periodic solutions bifurcate from the equilibrium point. Furthermore, the stability of these limit cycles was examined by analyzing the Jacobian matrix of the averaged equation. These results extend the applicability of averaging methods to high-order nonlinear differential equations and provide valuable insights into the dynamics and control of oscillatory phenomena in science and engineering.
- seventh-order differential equation,
- periodic solution,
- averaging theory,
- stability
Citation: Meriem Ladjimi. Existence and stability of limit cycles in a perturbed seventh-order differential equation[J]. AIMS Mathematics, 2025, 10(10): 24901-24922. doi: 10.3934/math.20251102

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Abstract

This paper investigates the existence and stability of limit cycles in a class of perturbed seventh-order non-autonomous differential equations that model multi-frequency oscillatory behavior with damping, commonly encountered in mechanical and control systems. The high order of the equation presents significant challenges in detecting periodic solutions and analyzing their stability. By applying first-order averaging theory, we reduced the original equation to a lower-dimensional averaged equation. We then derived explicit conditions under which isolated periodic solutions bifurcate from the equilibrium point. Furthermore, the stability of these limit cycles was examined by analyzing the Jacobian matrix of the averaged equation. These results extend the applicability of averaging methods to high-order nonlinear differential equations and provide valuable insights into the dynamics and control of oscillatory phenomena in science and engineering.

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