Periodic solutions and limit cycles of mixed Lienard-type differential equations

K. K. D. Adjaï; J. Akande; A. V. R. Yehossou; M. D. Monsia; K. K. D. Adjaï; J. Akande; A. V. R. Yehossou; M. D. Monsia

doi:10.3934/math.2022833

AIMS Mathematics

2022, Volume 7, Issue 8: 15195-15211. doi: 10.3934/math.2022833

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Research article

Periodic solutions and limit cycles of mixed Lienard-type differential equations

Department of Physics, University of Abomey-Calavi, Abomey-Calavi, 01.BP.526, Cotonou, Benin

Received: 29 March 2022 Revised: 25 May 2022 Accepted: 02 June 2022 Published: 16 June 2022
MSC : 34A05, 34C05, 34C25, 34C15

In the attractive research field of nonlinear differential equations, there are a few studies devoted to finding exact and explicit harmonic and isochronous periodic solutions and limit cycles. In this contribution, we present some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions. These classes of equations constitute counterexamples of the classical existence theorems.
Citation: K. K. D. Adjaï, J. Akande, A. V. R. Yehossou, M. D. Monsia. Periodic solutions and limit cycles of mixed Lienard-type differential equations[J]. AIMS Mathematics, 2022, 7(8): 15195-15211. doi: 10.3934/math.2022833

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Abstract

In the attractive research field of nonlinear differential equations, there are a few studies devoted to finding exact and explicit harmonic and isochronous periodic solutions and limit cycles. In this contribution, we present some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions. These classes of equations constitute counterexamples of the classical existence theorems.

References

[1]	D. W. Jordan, P. Smith, Nonlinear ordinary differential equations: An introduction for scientists and engineers, New York: Oxford University press, 2007.
[2]	R. E. Mickens, Oscillations in planar dynamic systems, Vol. 37, Series on Advances in Mathematics for Applied Sciences, World Scientific, 1996.
[3]	S. Saha, G. Gangopadhyay, D. S. Ray, Reduction of kinetic equations to Lienard-Levinson-Smith form: Counting limit cycles, Int. J. Appl. Comput. Math., 5 (2019), 2–11. https://doi.org/10.1007/s40819-019-0628-9 doi: 10.1007/s40819-019-0628-9
[4]	S. Saha, G. Gangopadhyay, Where the Lienard-Levinson-Smith (LLS) theorem cannot be applied for a generalised Lienard system, arXiv, 2021. https://doi.org/10.48550/arXiv.2104.06043
[5]	R. Benterki, J. Llibre, Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory, J. Comput. Appl. Math., 313 (2016), 273–283. https://doi.org/10.1016/j.cam.2016.08.047 doi: 10.1016/j.cam.2016.08.047
[6]	N. Levinson, O. K. Smith, A general equation for relaxation oscillations, Duke Math. J., 9 (1942), 382–403. https://doi.org/10.1215/S0012-7094-42-00928-1 doi: 10.1215/S0012-7094-42-00928-1
[7]	G. Villari, F. Zanolin, On the qualitative behavior of a class of generalized lienard planar systems, J. Dyn. Differ. Equ., 34 (2021), 179–207. https://doi.org/10.1007/s10884-021-09984-2 doi: 10.1007/s10884-021-09984-2
[8]	K. K. D. Adjaï, J. Akande, M. Nonti, M. D. Monsia, Limit cycles of polynomial and nonpolynomial systems of differential equations, 2021.
[9]	K. K. D. Adjaï, J. Akande, M. Nonti, M. D. Monsia, Truly nonlinear oscillators with limit cycles and harmonic solutions, 2021.
[10]	J. Akande, K. K. D. Adjaï, A. V. R. Yehossou, M. D. Monsia, Limit cycles of truly nonlinear oscillator equations, 2021.
[11]	J. Akande, K. K. D. Adjaï, M. Nonti, M. D. Monsia, Counter-examples to the existence theorems of limit cycles of differential equations, 2021.
[12]	J. Akande, M. Nonti, K. K. D. Adjaï, M. D. Monsia, A modified hybrid Rayleigh-Van der Pol oscillator equation with exact harmonic solution, 2021.

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