Limit Cycles in Two Types of Symmetric Liénard Systems

Abstract

Liénard systems and their generalized forms are classical and important models of nonlinear oscillators, and have been widely studied by mathematicians and scientists. The main problem considered is the maximal number of limit cycles that the system can have. In this paper, two types of symmetric polynomial Liénard systems are investigated and the maximal number of limit cycles bifurcating from Hopf singularity is obtained. A global result is also presented.

Hopf bifurcation for two types of Liénard systems

“…When a 0 = a 1 = 0, a 2 = 81 52 , a 3 = − 54 13 , a 4 = 33 0 , a 1 , a 2 , a 4 , b 1 10 , and e 41 = 0, e 42 = e 43 = −45, e 44 = −12, e 45 = −24, e 46 = 8, e 47 = − 16 9 . Thus, as before, six limit cycles can appear near the origin.…”

“…The above theorem has many applications to various Liénard systems and certain models from biomathematics, see [7,9,10,20] and [21]. For example, Jiang et al [10] considered the systeṁ…”

Limit cycle bifurcations of a general Liénard system with polynomial restoring and damping functions

“…There are many examples in the natural sciences and technology in which this and related systems are applied (Rychkov, 1975;Lins et al, 1977;Lloyd, 1987;Bautin and Leontovich, 1990;Moreira, 1992;Smale, 1998;Gasull and Torregrosa, 1999;Agarwal and Ananthkrishnan, 2000;Owens et al, 2004;Jing et al, 2007;Slight et al, 2008). Such systems are often used to model either mechanical or electrical or biomedical systems, and in the literature many systems are transformed into Liénard type to aid in the investigations.…”

Cyclicity of some Liénard Systems

“…considered as an ideal in the ring of germs G θ * of real analytic functions at θ * (f k denotes the element induced by f k in G θ * ). More precisely, we need to find not any basis, but the basis constructed in the following recursive fashion: If we are able to find such a basis for the ideal I then following Bautin's approach we can rewrite (15) in the form (20). Then it is clear, that the multiplicity of 15is equal to s − 1, where s is the cardinality of the set M (see e.g.…”

“…This is an expression of the form (20), where the condition (19) holds near the point θ * . Therefore by Proposition 2.1 the multiplicity of F at θ * is at most s.…”