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 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 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 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 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 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 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

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 3-dimensional Hopf bifurcation via averaging theory

2007 ◽  
Vol 17 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Claudio A. Buzzi ◽  
Paulo R. da Silva ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
3 Dimensional
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