On the maximum number of limit cycles of a planar differential system

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria

Abstract

In this work, we are interested in the study of the limit cycles of a perturbed differential system in  \(\mathbb{R}^2\), given as follows
\[\left\{
\begin{array}{l}
\dot{x}=y, \\
\dot{y}=-x-\varepsilon (1+\sin ^{m}(\theta ))\psi (x,y),%
\end{array}%
\right.\]
where \(\varepsilon\) is small enough, \(m\) is a non-negative integer, \(\tan (\theta )=y/x\), and \(\psi (x,y)\) is a real polynomial of degree \(n\geq1\). We use the averaging theory of first-order to provide an upper bound for the maximum number of limit cycles. In the end, we present some numerical examples to illustrate the theoretical results.

Keywords

[1] S. Badi and A. Makhlouf, Limit cycles of the generalized Li´enard differential equation via averaging theory, Ann. Di. Eqs. 27(4) (2011) 472–479.
[2] I.S. Berezin and N.P. Zhidkv, Computing Methods, vol. II, Pergamon Press, Oxford, 1964.
[3] A. Buica, J. Gin´e and J. Llibre, Bifurcation of limit cycles from a polynomial degenerate center, Adv. Nonlinear Stud. 10 (2010) 597–609.
[4] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004) 7 22.
[5] T. Chen and J. Llibre, Limit cycles of a second-order differential equation, Appl. Math. Lett. 88 (2019) 111–117.
[6] A. Cima, J. Llibre, and M. A Teixeira, Limit cycles of some polynomial differential system in dimension 2, 3 and 4 via averaging theory, Appl. Anal. 87 (2008) 149–164.
[7] N. Hirano and S. Rybicki, Existence of limit cycles for coupled Van der Pol equations, J. Differ. Equ. 195(1) (2003) 194-209.
[8] J. Llibre and A.C. Mereu, Limit cycles for generalized Kukles polynomial differential systems, Nonlinear Anal. 74 (2011) 1261–1271.
[9] J. Llibre, R. Moeckel and C. Sim´o, Central Configurations, Periodic Orbits and Hamiltonian Systems, Advanced Courses in Mathematics CRM Barcelona, Birkh¨auser, 2015.
[10] N. Mellahi, A. Boulfoul and A. Makhlouf, Maximum number of limit cycles for generalized Kukles polynomial differential systems, Differ. Equ. Dyn. Syst. 27 (2019) 493-–514.
[11] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, V. 7, 3rd edition.Springer, New York, 2000.
 [12] H. Poincar´e, M´emoire sur les courbes d´efinies par une ´equation diff´erentielle I, II, J. Math. Pures Appl. 7 (1881) 375–422; 8 (1882) 251–296.
[13] E. S´aez and I. Szanto, Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse, Appl. Math. Lett. 25 (2012) 1695-–1700.
[14] H. Shi, Y. Bai and M. Han, On the maximum number of limit cycles for a piecewise smooth differential system, Bull. Sci. Math. 163 (2020) 102887.
[15] D. Zwillinger, Table of Integrals, Series and Products, 2014, ISBN: 978-0-12-384933-5.
Volume 13, Issue 1
March 2022
Pages 1462-1478
  • Receive Date: 04 April 2021
  • Accept Date: 12 September 2021