Global Hopf bifurcation analysis of a six-dimensional Fitzhugh-Nagumo neural network with delay by a synchronized scheme

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch

Research output: Contribution to journal › Article

1 Citation (Scopus)

Abstract

Global Hopf bifurcation analysis is carried out on a six-dimensional FitzHugh-Nagumo (FHN) neural network with a time delay. First, the existence of local Hopf bifurcations of the system is investigated and the explicit formulae which can determine the direction of the bifurcations and the stability of the periodic solutions are derived using the normal form method and the center manifold theory. Then the sufficient conditions for the system to have multiple periodic solutions when the delay is far away from the critical values of Hopf bifurcations are obtained by using the Wu's global Hopf bifurcation theory and the Bendixson's criterion. Especially, a synchronized scheme is used during the analysis to reduce the dimension of the system. Finally, example numerical simulations are given to support the theoretical analysis.

Original languageEnglish
Pages (from-to)457-474
Number of pages18
JournalDiscrete and Continuous Dynamical Systems - B
Volume16
Issue number2
DOIs
Publication statusPublished - Sep 2011

Fingerprint

FitzHugh-Nagumo
Hopf bifurcation
Global Bifurcation
Bifurcation Analysis
Hopf Bifurcation
Neural Networks
Neural networks
Periodic Solution
Local Bifurcations
Center Manifold
Bifurcation Theory
Normal Form
Critical value
Explicit Formula
Time Delay
Time delay
Theoretical Analysis
Bifurcation
Numerical Simulation
Sufficient Conditions

Keywords

  • Global Hopf bifurcation
  • FitzHugh-Nagumo model
  • neural network
  • delay
  • synchronize

Cite this

Global Hopf bifurcation analysis of a six-dimensional Fitzhugh-Nagumo neural network with delay by a synchronized scheme. / Han, Fang; Zhen, Bin; Du, Ying; Zheng, Yanhong; Wiercigroch, Marian.

In: Discrete and Continuous Dynamical Systems - B, Vol. 16, No. 2, 09.2011, p. 457-474.

Research output: Contribution to journal › Article

Han, F, Zhen, B, Du, Y, Zheng, Y & Wiercigroch, M 2011, 'Global Hopf bifurcation analysis of a six-dimensional Fitzhugh-Nagumo neural network with delay by a synchronized scheme', Discrete and Continuous Dynamical Systems - B, vol. 16, no. 2, pp. 457-474. https://doi.org/10.3934/dcdsb.2011.16.457
Han F, Zhen B, Du Y, Zheng Y, Wiercigroch M. Global Hopf bifurcation analysis of a six-dimensional Fitzhugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B. 2011 Sep;16(2):457-474. https://doi.org/10.3934/dcdsb.2011.16.457
Han, Fang ; Zhen, Bin ; Du, Ying ; Zheng, Yanhong ; Wiercigroch, Marian. / Global Hopf bifurcation analysis of a six-dimensional Fitzhugh-Nagumo neural network with delay by a synchronized scheme. In: Discrete and Continuous Dynamical Systems - B. 2011 ; Vol. 16, No. 2. pp. 457-474.
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