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.

Quasiperiodicity and Chaos Through Hopf–Hopf Bifurcation in Minimal Ring Neural Oscillators Due to a Single Shortcut

2019 ◽  
Vol 29 (05) ◽  
pp. 1950065
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima ◽  
Haruna Matsushita
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Neural Oscillator ◽  
Quasiperiodic Solution ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Neural Oscillators ◽  
Hopf Bifurcation Point ◽  
Codimension Two

Quasiperiodicity and chaos in a ring of unidirectionally coupled sigmoidal neurons (a ring neural oscillator) caused by a single shortcut is examined. A codimension-two Hopf–Hopf bifurcation for two periodic solutions exists in a ring of six neurons without self-couplings and in a ring of four neurons with self-couplings in the presence of a shortcut at specific locations. The locus of the Neimark–Sacker bifurcation of the periodic solution emanates from the Hopf–Hopf bifurcation point and a stable quasiperiodic solution is generated. Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation, and multiple chaotic oscillations are generated through period-doubling cascades of periodic solutions in the Arnold’s tongues. Further, such chaotic irregular oscillations due to a single shortcut are also observed in propagating oscillations in a ring of Bonhoeffer–van der Pol (BVP) neurons coupled unidirectionally by slow synapses.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

BIFURCATION STRUCTURE OF SCALAR DIFFERENTIAL DELAYED EQUATIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 657-665 ◽  
Author(s):  
C. P. MALTA ◽  
C. GROTTA RAGAZZO
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Negative Feedback ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Feedback Condition ◽  
Numerical Results ◽  
The Other ◽  
Bifurcation Structure ◽  
Dominant Frequencies

We study periodic solutions of the equation [Formula: see text], with f(X) given by f1(X) = AX(1 − X) or f2(X) = πµ (1 − sin X), grouped in some sets characterized by different dominant frequencies. Numerical results with f(X) = f1(X) are given. One of these sets is shown to exhibit period-doubling cascade in the direction of both parameters A and τ. The other sets are shown to exhibit many other period-doubling cascades as τ is varied establishing a relation between the bifurcation structure within the sets. Furthermore we obtain a lower bound on A and µ for the existence of more complex dynamics. We conjecture that this fact is related to the violation of the so-called "negative-feedback condition."

Periodic Orbits and Chaotic Sets in a Low-Dimensional Model for Shear Flows

2005 ◽  
Vol 4 (2) ◽  
pp. 352-376 ◽  
Author(s):  
Jeff Moehlis ◽  
Holger Faisst ◽  
Bruno Eckhardt
Keyword(s):  
Periodic Orbits ◽  
Shear Flows ◽  
Dimensional Model ◽  
Low Dimensional

scholarly journals The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

2021 ◽  
Author(s):  
Yehu Lv
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Gestation Delay ◽  
Predator Prey Model ◽  
Spatially Homogeneous ◽  
Type 3 ◽  
With Memory ◽  
Diffusive System

Abstract In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in a diffusive system with memory and general delays. The developed algorithm for calculating the normal form can be used to investigate the direction and stability of Hopf bifurcation. Then, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without considering gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. Furthermore, by using the developed algorithm for calculating the normal form, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found theoretically. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.

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.

COLLECTIVE DYNAMICS AND CONTROL OF A 3-D SMALL-WORLD NETWORK WITH TIME DELAYS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250281 ◽  
Author(s):  
XU XU ◽  
JIAWEI LUO ◽  
YUANTONG GU
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Periodic Solutions ◽  
Hybrid Control ◽  
Period Doubling ◽  
Small World ◽  
State Feedback Control ◽  
Small World Network ◽  
Delayed Systems ◽  
Collective Dynamics

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS I

2006 ◽  
Vol 16 (10) ◽  
pp. 2887-2902 ◽  
Author(s):  
ZHUJUN JING ◽  
JIANPING YANG
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Pendulum Equation ◽  
Parametric And External Excitations

Pendulum equation with parametric and external excitations is investigated in (I) and (II). In (I), by applying Melnikov's method, we prove the criterion of existence of chaos under periodic perturbation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors including the period-n (n = 2 to 6, 10, 15 and 20) orbits in different chaotic regions, interlocking periodic orbits, symmetry-breaking of periodic orbit, cascade of period-doubling bifurcations from period-5 and -10 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for a given external frequency or parametric frequency and chaos suddenly converting to periodic orbits, sudden jump in the size of attractors which is associated with the transverse intersection of stable and unstable manifolds of perturbed saddle, hopping behavior of chaos, transient chaos with complex periodic windows and interior crisis, varied chaotic attractors including the more than three-band and eight-band chaotic attractors, chaotic attractor after strange nonchaotic attractor. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping δ, spring constant α and frequency Ω of parametric excitation which can be considered as a control strategy. In (II), we will investigate the complex dynamics under quasi-periodic perturbation.

scholarly journals Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
L. K. Kana ◽  
A. Fomethe ◽  
H. B. Fotsin ◽  
E. T. Wembe ◽  
A. I. Moukengue
Keyword(s):  
Complex Dynamics ◽  
Spectral Characteristics ◽  
Period Doubling ◽  
Magnetic Coupling ◽  
Coupled System ◽  
Coupling Factor ◽  
Coupled Systems ◽  
Numerical Exploration ◽  
The Stability ◽  
Parameter Ranges

We propose the use of a simple, cheap, and easy technique for the study of dynamic and synchronization of the coupled systems: effects of the magnetic coupling on the dynamics and of synchronization of two Colpitts oscillators (wireless interaction). We derive a smooth mathematical model to describe the dynamic system. The stability of the equilibrium states is investigated. The coupled system exhibits spectral characteristics such as chaos and hyperchaos in some parameter ranges of the coupling. The numerical exploration of the dynamics system reveals various bifurcations scenarios including period-doubling and interior crisis transitions to chaos. Moreover, various interesting dynamical phenomena such as transient chaos, coexistence of solution, and multistability (hysteresis) are observed when the magnetic coupling factor varies. Theoretical reasons for such phenomena are provided and experimentally confirmed with practical measurements in a wireless transfer.

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