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Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop
TLDR
In this paper, the authors studied limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin.Abstract:
In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclnic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.read more
Citations
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Journal ArticleDOI
Bifurcation of limit cycles in piecewise smooth systems via Melnikov function
Maoan Han,Lijuan Sheng +1 more
TL;DR: In this article, the role of the rstorder Melnikov functions in studying the number of limit cycles of piecewisesmooth near-Hamiltonian systems on the plane is discussed.
Journal ArticleDOI
Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation
TL;DR: A lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line is provided.
Journal ArticleDOI
Asymptotic expansions of melnikov functions and limit cycle bifurcations
TL;DR: A series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions of the Melnikov function at these values are presented.
Journal ArticleDOI
A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system
Shimin Li,Changjian Liu +1 more
TL;DR: In this paper, a class of planar piecewise smooth quadratic integrable non-Hamiltonian systems with a center was studied and an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n was given.
Journal ArticleDOI
Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems
Feng Liang,Maoan Han,Xiang Zhang +2 more
TL;DR: In this paper, the authors studied the cyclicity of a generalized homoclinic loop of a piecewise smooth differential system and proved the existence of one or two limit cycles which are bifurcated from it.
References
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Book
Differential Equations with Discontinuous Righthand Sides
TL;DR: The kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics, algebraic geometry interacts with physics, and such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes.
Book
Theory of Oscillators
A. A. (Aleksandr Aleksandrovich) Andronov,S. Ė. Khaĭkin,Vitt, A. A. , d.,F. Immirzi,Wilfred Fishwick +4 more
Book
Non-Smooth Dynamical Systems
TL;DR: In this paper, the application of Conley index theory to non-smooth dynamical systems is discussed, and a general theory of differential inclusions is proposed. But it is not discussed in detail.
Journal ArticleDOI
On Hopf bifurcation in non-smooth planar systems
Maoan Han,Weinian Zhang +1 more
TL;DR: In this article, the Hopf bifurcation problem for non-smooth planar systems was studied and it was shown that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for FF or FP type and order 2 for PP type).
Journal ArticleDOI
Bifurcation of Limit Cycles by Perturbing Piecewise Hamiltonian Systems
TL;DR: An expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits of piecewise Hamiltonian systems on the plane.
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