scholarly journals On the Generalized Reflective Function of the Three-Degree Polynomial Differential System

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Reflective Function ◽  
Degree Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Existence Of Periodic Solutions

The structure of the generalized reflective function of three-degree polynomial differential systems is considered in this paper. The generated results are used for discussing the existence of periodic solutions of these systems.

scholarly journals Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Amar Makhlouf ◽  
Lilia Bousbiat
Keyword(s):  
Small Parameter ◽  
Real Number ◽  
Periodic Solutions ◽  
Differential System ◽  
Sufficient Conditions ◽  
Periodic Functions ◽  
Third Order ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Existence Of Periodic Solutions

We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential systemx.=-y+εP(x,y,z)+h1(t),  y.=x+εQ(x,y,z)+h2(t),  and  z.=az+εR(x,y,z)+h3(t), whereP,Q, andRare polynomials in the variablesx,y, andzof degreen,  hi(t)=hi(t+2π)withi=1,2,3being periodic functions,ais a real number, andεis a small parameter.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals Periodic Orbits Bifurcating from a Nonisolated Zero–Hopf Equilibrium of Three-Dimensional Differential Systems Revisited

2018 ◽  
Vol 28 (05) ◽  
pp. 1850058 ◽  
Author(s):  
Murilo R. Cândido ◽  
Jaume Llibre
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Three Dimensional ◽  
Averaging Theory ◽  
Chaotic Attractors ◽  
Differential Systems ◽  
Polynomial Differential System

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in [Formula: see text]. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in [Formula: see text] having [Formula: see text]-scroll chaotic attractors.

scholarly journals On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550131 ◽  
Author(s):  
Fangfang Jiang ◽  
Junping Shi ◽  
Jitao Sun
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Theory ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = -y + g1(x) + f1(x)y and y′ = x + g2(x) + f2(x)y, where fi(x) and gi(x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

scholarly journals Bifurcations of the Riccati Quadratic Polynomial Differential Systems

2021 ◽  
Vol 31 (06) ◽  
pp. 2150094
Author(s):  
Jaume Llibre ◽  
Bruno D. Lopes ◽  
Paulo R. da Silva
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Topological Equivalence ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Global Phase Portrait ◽  
Poincaré Disk

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

scholarly journals Existence for Singular Periodic Problems: A Survey of Recent Results

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Jifeng Chu ◽  
Juntao Sun ◽  
Patricia J. Y. Wong
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Impulsive Differential Equations ◽  
Differential Systems ◽  
Singular Differential Equations ◽  
Periodic Problems ◽  
Existence Of Periodic Solutions

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

scholarly journals Periodic Solutions of a Periodic FitzHugh–Nagumo System

2015 ◽  
Vol 25 (13) ◽  
pp. 1550180 ◽  
Author(s):  
Jaume Llibre ◽  
Claudio Vidal
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Existence Of Periodic Solutions

Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.

scholarly journals ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF NONLINEAR DIFFERENTIAL SYSTEMS

1993 ◽  
Vol 24 (2) ◽  
pp. 173-188
Author(s):  
LIHONG HUANG ◽  
JIANSHE YU
Keyword(s):  
Periodic Solutions ◽  
Special Form ◽  
Differential Systems ◽  
Nontrivial Periodic Solutions ◽  
Existence Of Periodic Solutions ◽  
Nonlinear Differential Systems

In this paper three theorems on the existence of nontrivial periodic solutions of the system \[ dx/dt =e(y)\]\[dy/dt =-e(y)f(x)- g(x)\] are proved, which not only generalize some known results on the existence of periodic solutions of Lienard's system (i.e. the special form for $e(y) = y$), but also relax or eliminate some traditional assumptions.

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