Skip to main content
SpringerLink
Log in

Hopf Bifurcation Analysis of a Reaction-Diffusion Neural Network with Time Delay in Leakage Terms and Distributed Delays

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

In this paper, a reaction-diffusion neural network with time delay in leakage terms and distributed synaptic transmission delays under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation are established. By using the normal form theory and the center manifold reduction of partial functional differential equations, explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Boushaba K, Ruan S (2001) Instability in diffusive ecological models with nonlocal delay effects. J Math Anal Appl 258:269–286

    Article  MathSciNet  MATH  Google Scholar 

  2. Chua LO (1999) Passivity and complexity. IEEE Trans Circuit Syst I 46:71–82

    Article  MathSciNet  MATH  Google Scholar 

  3. Faria T (2000) On a planar system modelling a neuron network with memory. J Differ Equ 168:129–149

    Article  MathSciNet  MATH  Google Scholar 

  4. Gopalsamy K (2007) Leakage delays in BAM. J Math Anal Appl 325:1117–1132

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo S, Huang L (2003) Periodic solutions in an inhibitory two-neuron network. J Comput Appl Math 161:217–229

    Article  MathSciNet  MATH  Google Scholar 

  6. Hale J (1977) Theory of functional differential equations. Springer, Heidelberg

    Book  MATH  Google Scholar 

  7. Haykin S (1999) Neural networks. Prentice Hall, New Jersey

    MATH  Google Scholar 

  8. Kosko B (1992) Neural networks and fuzzy systems. Prentice Hall, New Delhi

    MATH  Google Scholar 

  9. Liao X, Wong KW, Wu Z (2001) Bifurcation analysis on a two-neuron system with distributed delays. Phys D 149:123–141

    Article  MathSciNet  MATH  Google Scholar 

  10. Liao X, Li S, Wong KW (2003) Hopf bifurcation on a two-neuron system with distributed delays: a frequency domain approach. Nonlinear Dyn 31:299–326

    Article  MathSciNet  MATH  Google Scholar 

  11. Olien L, Belair J (1997) Bifurcation, stability and monotonicity properties of a delayed neural model. Phys D 102:349–363

    Article  MathSciNet  MATH  Google Scholar 

  12. Peng S (2010) Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms. Nonlinear Anal RWA 11:2141–2151

    Article  MathSciNet  MATH  Google Scholar 

  13. Ruan S (1998) Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling. IMA J Appl Math 61:15–32

    Article  MathSciNet  MATH  Google Scholar 

  14. Ruan S, Filfil RS (2004) Dynamics of a two-neuron system with discrete and distributed delays. Phys D 191:323–342

    Article  MathSciNet  MATH  Google Scholar 

  15. Tu F, Liao X, Zhang W (2006) Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos Solitons Fractals 28:437–447

    Article  MathSciNet  MATH  Google Scholar 

  16. Wei J, Ruan S (1999) Stability and bifurcation in a neural network model with two delays. Phys D 130:255–272

    Article  MathSciNet  MATH  Google Scholar 

  17. Wu J (1996) Theory and applications of partial functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  18. Yuan Z, Huang L (2003) Convergence and periodicity in a discrete-time network of two neurons with self-connections. Comput Math Appl 46:1337–1345

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhao H (2004) Global asymptotic stability of Hopfield neural network involving distributed delays. Neural Netw 17:47–53

    Article  MATH  Google Scholar 

  20. Zhou S, Liao X, Yu J et al (2004) Chaos and its synchronization in two-neuron systems with discrete delays. Chaos Solitons Fractals 21:133–142

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhou X, Wu Y, Li Y et al (2009) Stability and Hopf bifurcation analysis on a two-neuron network with discrete and distributed delays. Chaos Solitons Fractals 40:1493–1505

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhu H, Huang L, Dai B (2004) Convergence and periodicity of solutions for a neural network of two neurons. Appl Math Comput 155:813–836

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11371368, 11071254, 61305076), the Natural Science Foundation of Hebei Province of China (No. A2014506015) and the Natural Science Foundation for Young Scientists of Hebei Province (No. A2013506012) and the Science Research Foundation of Shijiazhuang Mechanical Engineering College (Nos. YJJXM12010, YJJXM13008).

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang, 050003, Hebei, People’s Republic of China

    Xiaohong Tian & Rui Xu

Authors
  1. Xiaohong Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rui Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaohong Tian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, X., Xu, R. Hopf Bifurcation Analysis of a Reaction-Diffusion Neural Network with Time Delay in Leakage Terms and Distributed Delays. Neural Process Lett 43, 173–193 (2016). https://doi.org/10.1007/s11063-015-9410-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-015-9410-0

Keywords

Mathematics Subject Classification

Search

Navigation