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

Comments on “Shilnikov chaos and Hopf bifurcation in three-dimensional differential system”

Optik ◽  
2018 ◽  
Vol 155 ◽  
pp. 251-256 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Hopf Bifurcation ◽  
Differential System ◽  
Three Dimensional

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 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 Integrability and bifurcation of a three-dimensional circuit differential system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Brigita Ferčec ◽  
Valery G. Romanovski ◽  
Yilei Tang ◽  
Ling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Lyapunov Function ◽  
Differential System ◽  
Stable Equilibrium ◽  
Three Dimensional ◽  
Asymptotically Stable ◽  
Center Manifolds ◽  
Invariant Foliation ◽  
Invariant Surfaces

<p style='text-indent:20px;'>We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.</p>

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.

STABILITY AND HOPF BIFURCATION OF A DELAY DIFFERENTIAL SYSTEM IN MICROBIAL CONTINUOUS CULTURE

2009 ◽  
Vol 02 (03) ◽  
pp. 321-328 ◽  
Author(s):  
XIAOFANG LI ◽  
RONGNING QU ◽  
ENMIN FENG
Keyword(s):  
Hopf Bifurcation ◽  
Continuous Culture ◽  
Differential System ◽  
Center Manifold ◽  
Continuous Fermentation ◽  
Transversality Condition ◽  
Discrete Time Delay ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential

Introducing discrete time delay into the model for producing 1, 3-propanediol by microbial continuous fermentation, the stability and Hopf bifurcation of a delay differential system for microorganisms in continuous culture are considered in this paper, including the changing regularity of bifurcation value and oscillating period. Algebraic criteria for absolute stability, as well as the transversality condition for Hopf bifurcation of this kind system are obtained. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, numerical simulations show the effectiveness of our results. The pictures of periodic solutions and phase planes with specified parameters suggest that our results can qualitatively describe oscillatory phenomena occurring in experiments.

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.

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