New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

Safia Meftah, Fares Yazid, Tefaha Lejdel Ali, Keltoum Bouhali (), Abdelkader Moumen, Tariq Alraqad and Hicham Saber
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Safia Meftah: Laboratory of Operator Theory, EDP and Applications, Department of Mathematics, Science Exact Faculty, University of El Oued, El Oued 39000, Algeria
Fares Yazid: Laboratory of Pure and Applied Mathematics, Amar Telidji University of Laghouat, Laghouat 03000, Algeria
Tefaha Lejdel Ali: Laboratory of Operator Theory, EDP and Applications, Department of Mathematics, Science Exact Faculty, University of El Oued, El Oued 39000, Algeria
Keltoum Bouhali: Department of Mathematics, College of Sciences and Arts in Ar-Rass, Qassim University, Saudi Arabia
Abdelkader Moumen: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
Tariq Alraqad: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
Hicham Saber: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia

Mathematics, 2023, vol. 11, issue 23, 1-22

Abstract: A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

Keywords: asymptotic stability; Iterative methods; periodic solutions; Second Bogolubov’s theorem; Van der Pol equations; differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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