scholarly journals Zero-Hopf Periodic Orbits for a Rössler Differential System

2020 ◽  
Vol 30 (12) ◽  
pp. 2050170
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Differential System ◽  
Variable Formula ◽  
Independent Variable

We study the zero-Hopf bifurcation of the Rössler differential system [Formula: see text] where the dot denotes the derivative with respect to the independent variable [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] are real parameters.

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 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 Guaranteed Estimates for the Length of Branches of Periodic Orbits for Equivariant Hopf Bifurcation

2020 ◽  
Vol 30 (13) ◽  
pp. 2050198
Author(s):  
Edward Hooton ◽  
Zalman Balanov ◽  
Dmitrii Rachinskii
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Differential System ◽  
Bifurcation Point ◽  
Hopf Bifurcations ◽  
Linear Estimate ◽  
Hopf Bifurcation Point ◽  
Equivariant Degree ◽  
Equivariant Hopf Bifurcation

Connected branches of periodic orbits originating at a Hopf bifurcation point of a differential system are considered. A computable estimate for the range of amplitudes of periodic orbits contained in the branch is provided under the assumption that the nonlinear terms satisfy a linear estimate in a ball. If the estimate is global, then the branch is unbounded. The results are formulated in an equivariant setting where the system can have multiple branches of periodic orbits characterized by different groups of symmetries. The nonlocal analysis is based on the equivariant degree method, which allows us to handle both generic and degenerate Hopf bifurcations. This is illustrated by examples.

scholarly journals On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system

2015 ◽  
Vol 80 (1-2) ◽  
pp. 353-361 ◽  
Author(s):  
Jaume Llibre ◽  
Regilene D. S. Oliveira ◽  
Claudia Valls
Keyword(s):  
Hopf Bifurcation ◽  
Differential System

Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

2017 ◽  
Vol 27 (07) ◽  
pp. 1750105 ◽  
Author(s):  
Shuling Yan ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Volterra Model ◽  
Delay Effect ◽  
Cross Diffusion ◽  
Bifurcation Phenomena ◽  
Bifurcation Direction ◽  
Steady State Solutions

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

scholarly journals Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

2004 ◽  
Vol 90 (1-2) ◽  
pp. 87-107 ◽  
Author(s):  
Mercè Ollé ◽  
Joan R. Pacha ◽  
Jordi Villanueva
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits

Analysis of Zero-Hopf Bifurcation in Two Rössler Systems Using Normal Form Theory

2020 ◽  
Vol 30 (16) ◽  
pp. 2030050
Author(s):  
Bing Zeng ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Orbits ◽  
Critical Points ◽  
Normal Forms ◽  
Averaging Method ◽  
Hopf Bifurcations ◽  
Time Averaging ◽  
Normal Form Theory ◽  
Form Theory

In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation.

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