The Dynamics of a Kind of Liénard System with Sixth Degree and Its Limit Cycle Bifurcations Under Perturbations

In this paper, the different topological types of phase portrait of the unperturbed Liénard system x ˙ = y , y ˙ = - g ( x ) are given, where deg g ( x ) = 6 . We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic l...

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Bibliographic Details
Published in:Qualitative theory of dynamical systems 2020-04, Vol.19 (1), Article 26
Main Authors: Han, Maoan, Yang, Junmin
Format: Article
Language:English
Subjects:
Bifurcations
Difference and Functional Equations
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
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Summary:In this paper, the different topological types of phase portrait of the unperturbed Liénard system x ˙ = y , y ˙ = - g ( x ) are given, where deg g ( x ) = 6 . We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, we give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, we present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a kind of Liénard system under perturbations.
ISSN:1575-5460
1662-3592
DOI:10.1007/s12346-020-00377-2