scholarly journals Integrability and a Limit Cycle Solver for A Generalization of Polynomial Liénard Differential Systems

2018 ◽  
Vol 12 (2) ◽  
pp. 41-47
Author(s):  
Fatima FENENICHE ◽  
Med-Salem REZAOUI
Keyword(s):  
Limit Cycle ◽  
Differential Systems

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

scholarly journals LIMIT CYCLES FOR A CLASS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH THREE ZONES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250138 ◽  
Author(s):  
MAURÍCIO FIRMINO SILVA LIMA ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Vector Fields ◽  
Poincaré Map ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Vector ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Unique Limit

In this paper, we consider a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, we show that these systems admit always a unique limit cycle, which is hyperbolic.

scholarly journals Asymptotic Expansion of the Heteroclinic Bifurcation for the Planar Normal Form of the 1:2 Resonance

2016 ◽  
Vol 26 (01) ◽  
pp. 1650017 ◽  
Author(s):  
Luci A. F. Roberto ◽  
Paulo R. da Silva ◽  
Joan Torregrosa
Keyword(s):  
Asymptotic Expansion ◽  
Normal Form ◽  
Phase Portrait ◽  
Limit Cycle ◽  
Heteroclinic Bifurcation ◽  
Differential Systems ◽  
The Family ◽  
Heteroclinic Connection ◽  
Parameter Values

We consider the family of planar differential systems depending on two real parameters [Formula: see text] This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve [Formula: see text] [Formula: see text] We significantly improve the knowledge of this curve in a neighborhood of the origin.

Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

2020 ◽  
Vol 30 (03) ◽  
pp. 2050042
Author(s):  
Jihua Yang
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Isochronous Systems

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

scholarly journals Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

2018 ◽  
Vol 61 (2) ◽  
pp. 499-512 ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Algebraic Curve ◽  
Positive Answer ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

scholarly journals On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones

2015 ◽  
Vol 25 (04) ◽  
pp. 1550059 ◽  
Author(s):  
Maurício Firmino Silva Lima ◽  
Claudio Pessoa ◽  
Weber F. Pereira
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Vector Fields ◽  
Poincaré Map ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Vector ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Unique Limit

Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.

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