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.

Fold-Pitchfork Bifurcation, Arnold Tongues and Multiple Chaotic Attractors in a Minimal Network of Three Sigmoidal Neurons

2018 ◽  
Vol 28 (10) ◽  
pp. 1850123 ◽  
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima ◽  
Haruna Matsushita
Keyword(s):  
Periodic Solutions ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Quasiperiodic Solution ◽  
Chaotic Attractors ◽  
Dimensional Parameter ◽  
Chaotic Neural Networks ◽  
Minimal Network ◽  
Arnold Tongues ◽  
Period Doubling Bifurcations

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

2019 ◽  
Vol 29 (11) ◽  
pp. 1950154 ◽  
Author(s):  
Jiazhe Lin ◽  
Rui Xu ◽  
Xiaohong Tian
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Codimension Two ◽  
Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

scholarly journals Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
J. Xu ◽  
K. W. Chung
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Efficient Method ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
Delayed Systems ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
System With Time Delay

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

2017 ◽  
Vol 78 (4) ◽  
pp. 1217-1231 ◽  
Author(s):  
Jifeng Cui ◽  
Wenyu Zhang ◽  
Zeng Liu ◽  
Jianglong Sun
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
Period Doubling ◽  
Van Der Pol

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.

ON THE NUMERICAL DETECTION OF CODIMENSION-3 BIFURCATIONS OF PERIODIC SOLUTIONS (WITH APPLICATION TO OPTICAL BISTABILITY)

1994 ◽  
Vol 04 (06) ◽  
pp. 1425-1446
Author(s):  
KLAUS-GEORG NOLTE ◽  
IVAN L’HEUREUX
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Optical Bistability ◽  
Continuation Method ◽  
Period Doubling ◽  
Bistable System ◽  
Dimensional System ◽  
Systems Of Equations ◽  
The Stability ◽  
Stationary Behavior

Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.

scholarly journals All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Sorinel A. Oprisan
Keyword(s):  
Hopf Bifurcation ◽  
Oscillatory Behavior ◽  
Positive Phase ◽  
Neural Oscillator ◽  
Phase Resetting ◽  
Invariant Circle ◽  
Neural Oscillators ◽  
External Stimuli ◽  
Firing Period ◽  
Phase Resetting Curves

Phase resetting curves (PRCs) are phenomenological and quantitative tools that tabulate the transient changes in the firing period of endogenous neural oscillators as a result of external stimuli, for example, presynaptic inputs. A brief current perturbation can produce either a delay (positive phase resetting) or an advance (negative phase resetting) of the subsequent spike, depending on the timing of the stimulus. We showed that any planar neural oscillator has two remarkable points, which we called neutral points, where brief current perturbations produce no phase resetting and where the PRC flips its sign. Since there are only two neutral points, all PRCs of planar neural oscillators are bimodal. The degree of bimodality of a PRC, that is, the ratio between the amplitudes of the delay and advance lobes of a PRC, can be smoothly adjusted when the bifurcation scenario leading to stable oscillatory behavior combines a saddle node of invariant circle (SNIC) and an Andronov-Hopf bifurcation (HB).

FOOD WEB CHAOS WITHOUT SUBCHAIN OSCILLATORS

2005 ◽  
Vol 15 (11) ◽  
pp. 3481-3492 ◽  
Author(s):  
BRIAN BOCKELMAN ◽  
BO DENG
Keyword(s):  
Hopf Bifurcation ◽  
Singular Perturbation ◽  
Food Web ◽  
Bifurcation Point ◽  
Period Doubling ◽  
Reproductive Rate ◽  
Hopf Bifurcation Point ◽  
Parameter Values ◽  
The Web

A basic food web of four species is considered, of which there is a bottom prey X, two competing predators Y, Z on X, and a super predator W only on Y. The main finding is that population chaos does not require the existence of oscillators in any subsystem of the web. This minimum population chaos is demonstrated by increasing the relative reproductive rate of Z alone without alternating any other parameter nor any nullcline of the system. It occurs as the result of a period-doubling cascade from a Hopf bifurcation point. The method of singular perturbation is used to determine the Hopf bifurcation involved as well as the parameter values.

Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

1992 ◽  
Vol 340 (1659) ◽  
pp. 473-501 ◽  
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Phase Locking ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Wide Range

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

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