scholarly journals LIMIT CYCLE BIFURCATIONS IN A CLASS OF PIECEWISE SMOOTH DIFFERENTIAL SYSTEMS UNDER NON-SMOOTH PERTURBATIONS

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Jihua Yang ◽  
Keyword(s):  
Limit Cycle ◽  
Piecewise Smooth ◽  
Differential Systems

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

scholarly journals A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle

2019 ◽  
pp. 285-297 ◽  
Author(s):  
Ahmed Bendjeddou ◽  
Mohamed Grazem
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Polar Coordinates ◽  
Differential Systems ◽  
Ecological Economic ◽  
Quartic Polynomials ◽  
Algebraic Limit ◽  
Research Studies

Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Sub- sequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form x=xP4 (x;y); y= y Q4 (x; y) ; where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant {(x; y) 2 R2; x > 0; y > 0}. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result

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