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Nonlinear Dynamics

, Volume 78, Issue 3, pp 2085–2099 | Cite as

Stability analysis and rich oscillation patterns in discrete-time FitzHugh–Nagumo excitable systems with delayed coupling

  • Xiujuan Wang
  • Mingshu Peng
  • Ranran Cheng
  • Jinchen Yu
Original Paper

Abstract

In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.

Keywords

Neural network models Delay Stability Bifurcations Synchronization/asynchronization Chaos 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Xiujuan Wang
    • 1
    • 2
  • Mingshu Peng
    • 1
  • Ranran Cheng
    • 1
  • Jinchen Yu
    • 1
    • 3
  1. 1.Department of MathematicsBeijing Jiao Tong UniversityBeijing People’s Republic China
  2. 2.Weifang UniversityWeifang People’s Republic China
  3. 3.Shandong Jiaotong UniversityJinan People’s Republic China

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