Open AccessAveraging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
Author(s) -
Xingwu Chen,
Jaume Llibre,
Weinian Zhang
Publication year - 2017
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2017203
Subject(s) - mathematics , hopf bifurcation , quadratic differential , quadratic equation , quadratic function , limit cycle , bifurcation , polynomial , limit (mathematics) , mathematical analysis , differential (mechanical device) , plane (geometry) , center (category theory) , pure mathematics , geometry , physics , nonlinear system , chemistry , crystallography , quantum mechanics , thermodynamics
Agraïments: The first author is supported by NSFC grant #11471228. The third author is supported by NSFC grants #11231001, #11221101.It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear center at the origin of \R^2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \R^2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function