Four-dimensional Zero-Hopf Bifurcation of Quadratic Polynomial Differential System, via Averaging Theory of Third Order

2021 ◽  
Author(s):  
Djamila Djedid ◽  
El Ouahma Bendib ◽  
Amar Makhlouf
Keyword(s):  
Hopf Bifurcation ◽  
Differential System ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Third Order ◽  
Polynomial Differential System

scholarly journals On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1137
Author(s):  
Maoan Han ◽  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Equilibrium Points ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
3 Dimensional ◽  
Volterra Systems

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

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 Degenerate Fold–Hopf Bifurcations in a Rössler-Type System

2017 ◽  
Vol 27 (05) ◽  
pp. 1750068 ◽  
Author(s):  
G. Tigan ◽  
J. Llibre ◽  
L. Ciurdariu
Keyword(s):  
Hopf Bifurcation ◽  
Differential System ◽  
Type System ◽  
Averaging Theory ◽  
Hopf Bifurcations

We study the Hopf and the fold–Hopf bifurcations of the Rössler-type differential system [Formula: see text] with [Formula: see text]. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold–Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium [Formula: see text] when [Formula: see text]. This Hopf bifurcation becomes a fold–Hopf bifurcation when [Formula: see text].

scholarly journals Periodic Solutions of Continuous Third-Order Differential Equations with Piecewise Polynomial Nonlinearities

2020 ◽  
Vol 30 (11) ◽  
pp. 2050158
Author(s):  
J. Llibre ◽  
B. D. Lopes ◽  
J. R. de Moraes
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Differential System ◽  
Averaging Theory ◽  
Piecewise Polynomial ◽  
Third Order ◽  
Variable Formula

We consider third-order autonomous continuous piecewise differential equations in the variable [Formula: see text]. For such differential equations with nonlinearities of the form [Formula: see text], we investigate their periodic solutions using the averaging theory. We remark that since the differential system is only continuous we cannot apply to it the classical averaging theory, that needs that the differential system be at least of class [Formula: see text].

scholarly journals ON THE NUMBER OF LIMIT CYCLES FOR A GENERALIZATION OF LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350048 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLAUDIA VALLS
Keyword(s):  
Small Parameter ◽  
Periodic Orbits ◽  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
Polynomial Differential Systems

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

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 PERIODIC ORBITS NEAR EQUILIBRIA VIA AVERAGING THEORY OF SECOND ORDER

2012 ◽  
Vol 17 (5) ◽  
pp. 715-731
Author(s):  
Luis Barreira ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Periodic Orbits ◽  
Differential System ◽  
Sufficient Conditions ◽  
First Integral ◽  
Second Order ◽  
Averaging Theory ◽  
First Order

Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation.

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 3-dimensional Hopf bifurcation via averaging theory of third order

2017 ◽  
Vol 41 ◽  
pp. 1053-1071
Author(s):  
Elouahma BENDIB ◽  
Sabrina BADI ◽  
Amar MAKHLOUF
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
Third Order ◽  
3 Dimensional
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