On the periodic orbits bifurcating from a fold Hopf bifurcation in two hyperchaotic systems

Optik ◽  
2015 ◽  
Vol 126 (24) ◽  
pp. 4890-4895 ◽  
Author(s):  
A.E. Matouk
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Hyperchaotic Systems

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.

scholarly journals THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

2011 ◽  
Vol 21 (08) ◽  
pp. 2321-2330 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Transition Point ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System ◽  
Hamiltonian Hopf Bifurcation ◽  
Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

scholarly journals Bifurcation of an attracting invariant circle: a Denjoy attractor

1983 ◽  
Vol 3 (1) ◽  
pp. 87-118 ◽  
Author(s):  
Glen R. Hall
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Periodic Points ◽  
Invariant Circle ◽  
Worst Case ◽  
Dense Orbits ◽  
Parameter Values ◽  
Two Parameter

AbstractWe construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.

INTERMITTENCY AND CHAOS NEAR HOPF BIFURCATION WITH BROKEN 𝕆(2) × 𝕆(2) SYMMETRY

2013 ◽  
Vol 23 (08) ◽  
pp. 1350139
Author(s):  
YANG ZOU ◽  
GERHARD DANGELMAYR ◽  
IULIANA OPREA
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Normal Form ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Coupled System ◽  
Dimensional Model ◽  
Low Dimensional

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2) symmetry is perturbed by imperfection terms that break a continuous translation symmetry. The parameters of the fully symmetric normal form are fixed to values for which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is attracted by a chaotic attractor resulting from a period doubling cascade of periodic orbits. By using symmetry-adapted variables, the dimension of the phase space of the normal form is reduced to four and the dimension of the perturbed normal form is reduced to five. In the reduced phase space, periodic solutions are revealed as fixed points, and quasiperiodic solutions as periodic orbits. For the perturbed normal form, parameter regimes with different types of chaotic dynamics are identified when the imperfection parameter is varied. The characteristics of this complex dynamics are symmetry breaking and increasing, various period doubling cascades, intermittency and crises, and switching between symmetry-conjugated chaotic saddles. In particular, the perturbed system serves as a low dimensional model for the complicated switching dynamics found in simulations of the globally coupled system of Ginzburg–Landau equations extending the 𝕆(2) × 𝕆(2)-symmetric normal form to account for spatial modulations. In addition, this system can be considered as a low dimensional model for the dynamics of perturbed waves in anisotropic systems with imperfect geometries due to the presence of sidewalls.

HOPF BIFURCATION FROM LINES OF EQUILIBRIA WITHOUT PARAMETERS IN MEMRISTOR OSCILLATORS

2010 ◽  
Vol 20 (02) ◽  
pp. 437-450 ◽  
Author(s):  
MARCELO MESSIAS ◽  
CRISTIANE NESPOLI ◽  
VANESSA A. BOTTA
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Fixed Set ◽  
Electronic Element ◽  
Stable Periodic ◽  
Parameter Values

The memristor is supposed to be the fourth fundamental electronic element in addition to the well-known resistor, inductor and capacitor. Named as a contraction for memory resistor, its theoretical existence was postulated in 1971 by L. O. Chua, based on symmetrical and logical properties observed in some electronic circuits. On the other hand its physical realization was announced only recently in a paper published on May 2008 issue of Nature by a research team from Hewlett–Packard Company. In this work, we present the bifurcation analysis of two memristor oscillators mathematical models, given by three-dimensional five-parameter piecewise-linear and cubic systems of ordinary differential equations. We show that depending on the parameter values, the systems may present the coexistence of both infinitely many stable periodic orbits and stable equilibrium points. The periodic orbits arise from the change in local stability of equilibrium points on a line of equilibria, for a fixed set of parameter values. This phenomenon is a kind of Hopf bifurcation without parameters. We have numerical evidences that such stable periodic orbits form an invariant surface, which is an attractor of the systems solutions. The results obtained imply that even for a fixed set of parameters the two systems studied may or may not present oscillations, depending on the initial condition considered in the phase space. Moreover, when they exist, the amplitude of the oscillations also depends on the initial conditions.

scholarly journals Stability and Hopf bifurcation periodic orbits in delay coupled Lotka-Volterra ring system

2019 ◽  
Vol 17 (1) ◽  
pp. 962-978
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Ring System ◽  
Center Manifold Theory ◽  
Dihedral Groups ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory

Abstract In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.

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.

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