Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems
Emergent behaviors of the generalized Lohe matrix model
Parameter identification on Abelian integrals to achieve Chebyshev property
Xianbo Sun 1,2, , Zhanbo Chen 1, and Pei Yu 2,,
Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning 530003, China
Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7
* Corresponding author: pyu@uwo.ca
Fund Project: Pei Yu was supported by the Natural Sciences and Engineering Research Council of Canada (No. R2686A02). Xianbo Sun was partially supported by the Ontario Graduate Scholarship, NSFC(11861009), NSFGX(2018GXNSFAA138198) and Program for Innovative Team of GUFE(2018-2021)
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Chebyshev criterion is a powerful tool in the study of limit cycle bifurcations in dynamical systems based on Abelian integrals, but it is difficult when the Abelian integrals involve parameters. In this paper, we consider the Abelian integrals on the periodic annuli of a Hamiltonian with one parameter, arising from the generalized Liénard system, and identify the parameter values such that the Abelian integrals have Chebyshev property. In particular, the bounds on the number of zeros of the Abelian integrals are derived for different parameter intervals. The main mathematical tools are transformations and polynomial boundary theory, which overcome the difficulties in symbolic computations and analysis, arising from large parametric-semi-algebraic systems.
Keywords: Hilbert's 16th problem, Liénard system, limit cycle, heteroclinc loops, bifurcation.
Mathematics Subject Classification: Primary: 34C07, 34C10; Secondary: 34C23.
Citation: Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020375
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Table 1. Eight different topological phase eortraits of system (8)$ _{\epsilon = 0} $ with heteroclinic loops (red curves)
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Xianbo Sun Zhanbo Chen Pei Yu