Hopf bifurcation analysis of a four-neuron network with multiple time delays

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

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Bifurcation Analysis of Discrete-Time Systems with Multiple Time-Delays

Proceedings of the 2017 2nd International Conference on Electrical, Control and Automation Engineering (ECAE 2017) ◽

10.2991/ecae-17.2018.15 ◽

2018 ◽

Author(s):

Li Zeng

Keyword(s):

Discrete Time ◽

Bifurcation Analysis ◽

Time Delays ◽

Multiple Time ◽

Multiple Time Delays ◽

Discrete Time Systems ◽

Time Systems

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Impacts of Multiple Time Delays on a Gene Regulatory Network Mediated by Small Noncoding RNA

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500698 ◽

2020 ◽

Vol 30 (05) ◽

pp. 2050069

Author(s):

Ming Liu ◽

Fanwei Meng ◽

Dongpo Hu

Keyword(s):

Hopf Bifurcation ◽

Gene Regulatory Network ◽

Regulatory Network ◽

Time Delays ◽

Noncoding Rna ◽

Multiple Time ◽

Normal Form Theory ◽

Small Noncoding Rna ◽

Multiple Time Delays ◽

Gene Regulatory

In this paper, the impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA is studied. By analyzing the associated characteristic equation of the corresponding linearized system, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Furthermore, the explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are given by the center manifold theorem and the normal form theory for functional differential equations. Finally, some numerical simulations are demonstrated for supporting the theoretical results.

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Bifurcation analysis of two disc dynamos with viscous friction and multiple time delays

Applied Mathematics and Computation ◽

10.1016/j.amc.2018.10.090 ◽

2019 ◽

Vol 347 ◽

pp. 265-281 ◽

Cited By ~ 13

Author(s):

Zhouchao Wei ◽

Bin Zhu ◽

Jing Yang ◽

Matjaž Perc ◽

Mitja Slavinec

Keyword(s):

Bifurcation Analysis ◽

Time Delays ◽

Viscous Friction ◽

Multiple Time ◽

Multiple Time Delays

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Stability and Hopf Bifurcation Analysis in Hindmarsh–Rose Neuron Model with Multiple Time Delays

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741650187x ◽

2016 ◽

Vol 26 (11) ◽

pp. 1650187 ◽

Cited By ~ 7

Author(s):

Dongpo Hu ◽

Hongjun Cao

Keyword(s):

Hopf Bifurcation ◽

Numerical Simulations ◽

Time Delays ◽

Single Neuron ◽

Neuron Model ◽

Multiple Time ◽

Bifurcation Diagrams ◽

Dynamical Behaviors ◽

Multiple Time Delays ◽

The Stability

In this paper, the dynamical behaviors of a single Hindmarsh–Rose neuron model with multiple time delays are investigated. By linearizing the system at equilibria and analyzing the associated characteristic equation, the conditions for local stability and the existence of local Hopf bifurcation are obtained. To discuss the properties of Hopf bifurcation, we derive explicit formulas to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions occurring through Hopf bifurcation. The qualitative analyses have demonstrated that the values of multiple time delays can affect the stability of equilibrium and play an important role in determining the properties of Hopf bifurcation. Some numerical simulations are given for confirming the qualitative results. Numerical simulations on the effect of delays show that the delays have different scales when the two delay values are not equal. The physiological basis is most likely that Hindmarsh–Rose neuron model has two different time scales. Finally, the bifurcation diagrams of inter-spike intervals of the single Hindmarsh–Rose neuron model are presented. These bifurcation diagrams show the existence of complex bifurcation structures and further indicate that the multiple time delays are very important parameters in determining the dynamical behaviors of the single neuron. Therefore, these results in this paper could be helpful for further understanding the role of multiple time delays in the information transmission and processing of a single neuron.

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Global Stability, Bifurcation, and Chaos Control in a Delayed Neural Network Model

Advances in Artificial Neural Systems ◽

10.1155/2014/369230 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Amitava Kundu ◽

Pritha Das

Keyword(s):

Neural Network ◽

Network Model ◽

Neural Network Model ◽

Bifurcation Analysis ◽

Time Delays ◽

Chaos Control ◽

Multiple Time ◽

Multiple Time Delays ◽

The Neural Network

Conditions for the global asymptotic stability of delayed artificial neural network model of n (≥3) neurons have been derived. For bifurcation analysis with respect to delay we have considered the model with three neurons and used suitable transformation on multiple time delays to reduce it to a system with single delay. Bifurcation analysis is discussed with respect to single delay. Numerical simulations are presented to verify the analytical results. Using numerical simulation, the role of delay and neuronal gain parameter in changing the dynamics of the neural network model has been discussed.

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