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Multistability and Multiperiodicity for a Class of Cohen–Grossberg BAM Neural Networks with Discontinuous Activation Functions and Time Delays

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Abstract

In this paper, a general class of Cohen–Grossberg bidirectional associative memory neural networks (CGBAMNNs) with time-varying delays, distributed delays and discontinuous activation functions is investigated. By partitioning the state space, employing analysis approach and Cauchy convergence principle, sufficient conditions are established for the existence and local exponential stability of multiple equilibrium points, which ensure that \(2n\)-dimensional CGBAMNNs with \(k\)-level discontinuous activation functions can have \(k^n\) equilibrium points. As an extension of multistability, sufficient conditions are obtained to ensure the existence of \(k^n\) locally exponentially stable periodic orbits of the system when time-varying delays and external inputs are periodic. Finally, three examples are given to illustrate the feasibility and application of the obtained results.

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References

  1. Tian J, Zhong S (2011) Improved delay-dependent stability criterion for neural networks with time-varying delay. Appl Math Comput 217:10278–10288

    Article  MathSciNet  Google Scholar 

  2. Wang H, Song Q, Duan C (2010) LMI criteria on exponential stability of BAM neural networks with both time-varying delays and general activation functions. Math Comput Simul 81:837–850

    Article  MathSciNet  Google Scholar 

  3. Zhang X, Wu S, Li K (2011) Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Commun Nonlinear Sci Numer Simul 16:1524–1532

    Article  MathSciNet  Google Scholar 

  4. Fu X, Li X (2011) LMI conditions for stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays. Commun Nonlinear Sci Numer Simulat 16:435–454

    Article  MathSciNet  Google Scholar 

  5. Zhou B, Song Q, Wang H (2011) Global exponential stability of neural networks with discrete and distributed delays and general activation functions on time scales. Neurocomputing 74:3142–3150

    Article  Google Scholar 

  6. Pan J, Zhan Y (2011) On periodic solutions to a class of non-autonomously delayed reaction-diffusion neural networks. Commun Nonlinear Sci Numer Simul 16:414–422

    Article  MathSciNet  Google Scholar 

  7. Huang Z, Mohamod S, Bin H (2010) Multiperiodicity analysis and numerical simulation of discrete-time transiently chaotic non-autonomous neural networks with time-varying delays. Commun Nonlinear Sci Numer Simul 15:1348–1357

    Article  MathSciNet  Google Scholar 

  8. Wang L, Lu W, Chen T (2009) Multistability and new attraction basins of almost-periodic solutions of delayed neural networks. IEEE Trans Neural Netw 20:1581–1593

    Article  Google Scholar 

  9. Kaslik E, Sivasundaram S (2011) Multiple periodic solutions in impulsive hybrid neural networks with delays. Appl Math Comput 217:4890–4899

    Article  MathSciNet  Google Scholar 

  10. Kaslik E, Sivasundaram S (2011) Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis. Neural Netw 24:370–377

    Article  Google Scholar 

  11. Kaslik E, Sivasundaram S (2011) Multistability in impulsive hybrid Hopfield neural networks with distributed delays. Nonlinear Anal Real World Appl 12:1640–1649

    Article  MathSciNet  Google Scholar 

  12. Zhou T, Wang M, Long M (2012) Existence and exponential stability of multiple periodic solutions for a multidirectional associative memory neural network. Neural Process Lett 35:187–202

    Article  Google Scholar 

  13. Huang Y, Zhang X (2011) Multistability properties of linear threshold discrete-time recurrent neural networks. Int J Inf Syst Sci 7:1–10

    MathSciNet  Google Scholar 

  14. Cao J, Feng G, Wang Y (2008) Multistability and multiperiodicity of delayed Cohen-Grossberg neural networks with a general class of activation functions. Physica D 237:1734–1749

    Article  MathSciNet  Google Scholar 

  15. Huang Z, Feng C, Mohamad S (2012) Multistability analysis for a general class of delayed Cohen-Grossberg neural networks. Inf Sci 187:233–244

    Article  MathSciNet  Google Scholar 

  16. Nie X, Huang Z (2012) Multistability and multiperiodicity of hign-order competitive neural networks with a general calss of activation functions. Neurocomputing 82:1–13

    Article  Google Scholar 

  17. Huang G, Cao J (2008) Multistability in bidirectional associative memory neural networks. Phys Lett A 372:2842–2854

    Article  MATH  Google Scholar 

  18. Huang G, Cao J (2010) Delay-dependent multistability in recurrent neural networks. Neural Netw 23:201–209

    Article  Google Scholar 

  19. Nie X, Cao J (2011) Multistability of second-order competitive neural networks with nondecreasing saturated activation functions. IEEE Trans Neural Netw 22:1694–1708

    Article  MATH  Google Scholar 

  20. Ding W, Wang L (2011) \(2^N\) almost periodic attractors for Cohen-Grossberg-type BAM neural networks with variable coefficients and distributed delays. J Math Anal Appl 373:322–342

    Article  MathSciNet  Google Scholar 

  21. Cheng C, Lin K, Shih C (2007) Multistability and convergence in delayed neural networks. Physica D 225:61–74

    Article  MathSciNet  Google Scholar 

  22. Wang L, Chen T (2012) Multistability of neural networks with Mexican-hat-type activation functions. IEEE Trans Neural Netw Learn Syst 23:1816–1826

    Article  Google Scholar 

  23. Zhang L, Zhang Y, Yu J, Heng P (2009) Some multistability properties of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions. Neurocomputing 72:3809–3817

    Article  MATH  Google Scholar 

  24. Zeng Z, Huang T, Zheng W (2010) Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function. IEEE Trans Neural Netw 21:1371–1377

    Article  Google Scholar 

  25. Wang L, Lu W, Chen T (2010) Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions. Neural Netw 23:189–200

    Article  MATH  Google Scholar 

  26. Zeng Z, Zheng W (2012) Multistability of neural networks with time-varying delays and concave-convex characteristics. IEEE Neural Netw Learn Syst 23:293–305

    Article  MATH  Google Scholar 

  27. Forti M, Nistri P (2003) Global convergence of neural networks with discontinuous neuron activations. IEEE Trans Circuits Syst I 50:1421–1435

    Article  MathSciNet  Google Scholar 

  28. Huang G, Cao J (2008) Multistability of neural networks with discontinuous activation function. Commun Nonlinear Sci Numer Simul 13:2279–2289

    Article  MathSciNet  Google Scholar 

  29. Bao G, Zeng Z (2012) Analysis and design of associative memories based on recurrent neural network with discontinuous activation functions. Neurocomputing 77:101–107

    Article  MATH  Google Scholar 

  30. Huang Y, Zhang H, Wang Z (2012) Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions. Neurocomputing 91:21–28

    Article  Google Scholar 

  31. Huang Y, Zhang H, Wang Z (2012) Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions. Appl Math Comput 219:899–910

    Article  MathSciNet  MATH  Google Scholar 

  32. Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern 13:815–826

    Article  MathSciNet  MATH  Google Scholar 

  33. Kosko B (1987) Adaptive bidirectional associative memories. Appl Opt 26:4947–4960

    Article  Google Scholar 

  34. Kosko B (1988) Bidirectional associative memories. IEEE Trans Syst Man Cybernet 18:49–60

    Article  MathSciNet  Google Scholar 

  35. Filippov A (1988) Differential equations with discontinuous righthand sides. Mathematics and its applications (Soviet Series). Kluwer Academic, Boston, MA

    Book  Google Scholar 

  36. Forti M, Nistri P, Papini D (2005) Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain. IEEE Trans Neural Netw 16:1449–1463

    Article  MATH  Google Scholar 

  37. Zeng Z, Wang J (2009) Associative memories based on continuous-time cellular neural networks designed using space-invariant cloning templates. Neural Netw 22:651–657

    Article  Google Scholar 

  38. Zeng Z, Wang J (2007) Analysis and design of associative memories bases on recurrent neural networks with linear saturation activation functions and time-varying delays. Neural Comput 19:2149–2182

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371368, 61305076, 11071254) and the Natural Science Foundation of Hebei Province (A2013506012).

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Authors and Affiliations

  1. Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, 050003, People’s Republic of China

    Yanke Du & Rui Xu

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  1. Yanke Du
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  2. Rui Xu
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Correspondence to Yanke Du.

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Du, Y., Xu, R. Multistability and Multiperiodicity for a Class of Cohen–Grossberg BAM Neural Networks with Discontinuous Activation Functions and Time Delays. Neural Process Lett 42, 417–435 (2015). https://doi.org/10.1007/s11063-014-9364-7

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