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 Stability and Hopf Bifurcation Analysis of Coupled Optoelectronic Feedback Loops

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Gang Zhu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Center Manifold ◽  
Stable Equilibrium ◽  
Feedback Loops ◽  
Asymptotically Stable ◽  
Center Manifold Theorem ◽  
Feedback Strength ◽  
Coupling Parameters ◽  
Normal Form Method

The dynamics of a coupled optoelectronic feedback loops are investigated. Depending on the coupling parameters and the feedback strength, the system exhibits synchronized asymptotically stable equilibrium and Hopf bifurcation. Employing the center manifold theorem and normal form method introduced by Hassard et al. (1981), we give an algorithm for determining the Hopf bifurcation properties.

scholarly journals Author’s reply to: Comments on “Asymptotically stable equilibrium points in new chaotic systems”

Nova Scientia ◽  
2017 ◽  
Vol 9 (19) ◽  
pp. 906-909
Author(s):  
K. Casas-García ◽  
L. A. Quezada-Téllez ◽  
S. Carrillo-Moreno ◽  
J. J. Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Systems ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Linear Quadratic ◽  
The Monte Carlo Method

Since theorem 1 of (Elhadj and Sprott, 2012) is incorrect, some of the systems found in the article (Casas-García et al. 2016) may have homoclinic or heteroclinic orbits and may seem chaos in the Shilnikov sense. However, the fundamental contribution of our paper was to find ten simple, three-dimensional dynamic systems with non-linear quadratic terms that have an asymptotically stable equilibrium point and are chaotic, which was achieved. These were obtained using the Monte Carlo method applied specifically for the search of these systems.

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

BREAKDOWN OF LIBRATIONAL INVARIANT SURFACES

1999 ◽  
Vol 09 (05) ◽  
pp. 975-982
Author(s):  
MIQUEL ANGEL ANDREU ◽  
ALESSANDRA CELLETTI ◽  
CORRADO FALCOLINI
Keyword(s):  
Phase Space ◽  
Numerical Investigation ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
The Moon ◽  
Dimensional Phase Space ◽  
Invariant Surfaces ◽  
The Stability

A numerical investigation of the stability of invariant librational tori is presented. The method has been developed for a model describing the spin-orbit coupling in Celestial Mechanics. Periodic orbits approaching the librational torus are computed by means of Newton's method. According to Greene's criterion, their stability is strictly related to the survival of invariant tori. We consider librational tori around the main spin-orbit resonances (1:1, 3:2). Their existence provides the stability of the resonances, due to the confinement properties in the three-dimensional phase space associated to our model. The results are consistent with the actual observations of the eccentricity and of the oblateness parameter. A different behavior of the Moon and Mercury around the main resonances is evidenced, providing interesting suggestions about the different probabilities of capture in a resonance.

Shilnikov chaos and Hopf bifurcation in three-dimensional differential system

Optik ◽  
2016 ◽  
Vol 127 (19) ◽  
pp. 7648-7655 ◽  
Author(s):  
Qiong He ◽  
Hai-Yun Xiong
Keyword(s):  
Hopf Bifurcation ◽  
Differential System ◽  
Three Dimensional

scholarly journals Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 354
Author(s):  
Zouhair Diab ◽  
Juan L. G. Guirao ◽  
Juan A. Vera
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Differential System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Periodic Trajectories ◽  
First Order ◽  
Three Dimensional Space

The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive.

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.

Three-Dimensional Hopf Bifurcation for a Class of Cubic Kolmogorov Model

2014 ◽  
Vol 24 (03) ◽  
pp. 1450036 ◽  
Author(s):  
Chaoxiong Du ◽  
Qinlong Wang ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Three Dimensional ◽  
Singular Values ◽  
Bifurcation Problem ◽  
Disturbed Condition ◽  
Differential Systems ◽  
Kolmogorov Model ◽  
Polynomial Differential Systems ◽  
Fine Focus

We study the Hopf bifurcation for a class of three-dimensional cubic Kolmogorov model by making use of our method (i.e. singular values method). We show that the positive singular point (1, 1, 1) of an investigated model can become a fine focus of 5 order, and moreover, it can bifurcate five small limit cycles under certain coefficients with disturbed condition. In terms of three-dimensional cubic Kolmogorov model, published references can hardly be seen, and our results are new. At the same time, it is worth pointing out that our method is valid to study the Hopf bifurcation problem for other three-dimensional polynomial differential systems.

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