LIMIT CYCLES OF A CLASS OF PERTURBED DIFFERENTIAL SYSTEMS BIFURCATING FROM AN UNPERTURBED HAMILTONIAN CENTER

Amor Menaceur (), Yassine Bouattia (), Salem Alkhalaf and Asma Alharbi ()
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Amor Menaceur: Laboratory of ACED, Department of Mathematics, University of Guelma, P. O. Box 401, Guelma 24000, Algeria
Yassine Bouattia: Advanced Control laboratory (LABCAV), Department of Mathematics University of Guelma, P. O. Box 401 Guelma 24000, Algeria
Salem Alkhalaf: Department of Computer Engineering College of Computer, Qassim University Buraydah, Saudi Arabia
Asma Alharbi: Department of Mathematics College of Science, Qassim University Buraydah 51452, Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 08, 1-10

Abstract: We study the number of limit cycles of the following planar differential system: u̇ = v2a−1 − 𠜀(1 + Sn2m1ϕ)G1(u,v),v̇ = −v2b−1 − 𠜀(1 + Cs2m2ϕ)G2(u,v), where a,b,m1,m2 are positive integers, for every h = 1, 2, the polynomial Gh(u,v) has degree nh ≥ 2 with Gh(0, 0) = 0, Cs(ϕ) and Sn(ϕ) are generalized trigonometric functions and 𠜀 is a small parameter. We provide an accurate upper bound of the maximum number of limit cycles such that the above system can have bifurcation from an unperturbed Hamiltonian center, using the averaging theory of the first order. These findings contribute to the broader understanding of nonlinear dynamics and have potential applications in various fields such as control systems, signal processing, and artificial intelligence.

Keywords: Limit Cycle; Averaging Method; Hamiltonian System; Generalized Trigonometric Functions; Bifurcation; Nonlinear Equations; Artificial Intelligence (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25401450

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