scholarly journals Control by time delayed feedback near a Hopf bifurcation point

2017 ◽  
pp. 1-23
Author(s):  
Sjoerd Verduyn Lunel ◽  
Babette de Wolff
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Delayed Feedback ◽  
Hopf Bifurcation Point ◽  
Time Delayed Feedback

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.

Dynamics near a symmetric Hopf bifurcation

1988 ◽  
Vol 108 (3-4) ◽  
pp. 249-267 ◽  
Author(s):  
Wayne Nagata
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Symmetry Breaking ◽  
Bifurcation Point ◽  
Unperturbed System ◽  
System Of Differential Equations ◽  
Hopf Bifurcation Point ◽  
Secondary Bifurcations

SynopsisWe consider the effects of a small symmetry breaking perturbation on a system of differential equations near a Hopf bifurcation point, where the unperturbed system has O(2) symmetry. We show that there exist secondary bifurcations of invariant two-tori of solutions and that the flow on the tori can be quasiperiodic or weakly resonant (phase locked), depending on the size of the perturbation.

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

2011 ◽  
Vol 25 (29) ◽  
pp. 3977-3986 ◽  
Author(s):  
HUAGUANG GU ◽  
HUIMIN ZHANG ◽  
CHUNLING WEI ◽  
MINGHAO YANG ◽  
ZHIQIANG LIU ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Firing Pattern ◽  
Neuronal System ◽  
Neural Firing ◽  
Coherence Resonance ◽  
Saddle Node Bifurcation ◽  
Hopf Bifurcation Point ◽  
Firing Patterns ◽  
Saddle Node

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

ESTIMATION OF THE HOPF BIFURCATION POINT FOR AEROELASTIC SYSTEMS

2001 ◽  
Vol 248 (1) ◽  
pp. 31-42 ◽  
Author(s):  
A. SEDAGHAT ◽  
J.E. COOPER ◽  
A.Y.T. LEUNG ◽  
J.R. WRIGHT
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Hopf Bifurcation Point
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