Limit Cycle Bifurcations in Perturbations of a Reversible Quadratic System with a Non-rational First Integral
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Limit Cycle Bifurcations in Perturbations of a Reversible Quadratic System with a Non-rational First Integral Yanqin Xiong1 · Rong Cheng1 · Na Li2 Received: 5 January 2020 / Accepted: 20 October 2020 © Springer Nature Switzerland AG 2020
Abstract This paper investigates the Hopf cyclicity of a piecewise smooth quadratic polynomial system by Melnikov function method, whose unperturbed system is a concrete reversible quadratic system with a center at the origin and with a non-rational first integral. By comparing the obtained result for the piecewise case with the result for the smooth case, it shows that the piecewise system can have at least four more limit cycles around the origin than the smooth one. Keywords Hopf cyclicity · Limit cycle · Melnikov function · Piecewise quadratic polynomial system Mathematics Subject Classification 34C05
1 Introduction and Main Results The second part of the well-known Hilbert’s 16th problem asks for the maximum number and relative distributions of limit cycles of planar polynomial differential systems with a given degree (see [8]). There are many excellent articles on them. Please see [10,15,25] and references therein. In order to make this problem easier, many researchers have studied the maximum number of limit cycles on near-Hamiltonian polynomial or piecewise polynomial systems (see [9,16,23,24,26,29,30]).
The first author was supported by National Natural Science Foundation of China (11701289, 11501055) and Natural Science Foundation of Jiangsu Province (BK20170936).
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Yanqin Xiong [email protected]
1
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
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School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China 0123456789().: V,-vol
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It is easy to prove that near-Hamiltonian linear polynomial systems have no limit cycles. From [26], one can know that near-Hamiltonian piecewise linear polynomial systems can have 2 limit cycles. In [12,31], planar integrable quadratic systems with a center can be classified into four classes: Q 3H , Q 3R , Q 3L V and Q 4 . When they are perturbed inside quadratic polynomial systems, denoted by N the maximal number of limit cycles bifurcating from the periodic orbits. There are a lot of papers studying this number N , please see [2,4,12,17,18,21,31,32] and references therein. Following [14,22], for Q 3R , the quadratic systems can be written as x˙ = −y − ax 2 − y 2 , y˙ = x + bx y,
(1.1)
where a, b are constants. The author of the paper [14] proved that N = 2 for the case a + b = 1, b < 1, b = 23 . Gautier et al. [7] classified all reversible centers of genus one into 22 types and verified that N ≤ 3 for some types. Obviously, these quadratic systems are equivalent to near-Hamiltonian systems. Loud [20] classified all quadratic polynomial differential systems with an isochronous center into four classes: S1 , S2 , S3 , S4 . When they are perturbed inside quadratic polynomial
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