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Spatial disorder of Cellular Neural Networks

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Abstract

This paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a devil-staircase like function.

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References

  1. V.S. Afraimovich and S.-B. Hsu, Lectures on chaotic dynamical systems. National Tsing-Hua University, Hsinchu, Taiwan, 1998.

    Google Scholar 

  2. V.S. Afraimovich and S.-N. Chow, Topological spatial chaos and homoclinic points of Zd-actions in lattice dynamical systems. Japan J. Indust. Appl. Math.,12 (1995), 367–383.

    Article  MATH  MathSciNet  Google Scholar 

  3. V.S. Afraimovich and V.I. Nekorkin, Chaos of travelling waves in a discrete chain of diffusively coupled maps. Int. J. Bifur. and Chaos,4 (1994), 631–637.

    Article  MATH  MathSciNet  Google Scholar 

  4. V.S. Afraimovich, V.V. Bykov and L.P. Shil’nikov, On the structurally unstable attracting limit sets of the Lorentz attractor type. Trans. Mosc. Math. Soc.,2 (1993), 153–215.

    Google Scholar 

  5. J.C. Ban, C.-H. Hsu and S.-S. Lin, Devil staircase of gap maps. Int. J. Bifur. and Chaos, (2001), not published.

  6. V.N. Belykh, Chaotic and strange attractors of a two-dimensional map. Sbornik: Mathematics,186, No. 3 (1995), 311–326.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Bowen, Topological entropy for noncompact sets. Trans. Amer. Math. Soc.,184 (1973), 125–136.

    Article  MathSciNet  Google Scholar 

  8. J.W. Cahn, S.-N. Chow and E.S. Van Vleck, Spatially discrete nonlinear diffusion equations. Rocky Mount. J. Math.,25 (1995), 87–118.

    Article  MATH  Google Scholar 

  9. J.W. Cahn, J. Mallet-Paret and E.S. Van Vleck, Travelling wave solutions for systems of ODE’s on a two-dimensional spatial lattice. SIAM J. Appl. Math.,59 (1999), 455–493.

    MATH  MathSciNet  Google Scholar 

  10. S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, I. IEEE Trans. Circuits Syst.,42 (1995), 746–751.

    Article  MathSciNet  Google Scholar 

  11. S.-N. Chow, J. Mallet-Paret and E.S. Van Vleck, Dynamics of lattice differential equations. Int. J. Bifur. and Chaos,6 (1996), 1605–1621.

    Article  MATH  Google Scholar 

  12. S.-N. Chow, J. Mallet-Paret and E.S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations. Random and Comp. Dyn.,4 (1996), 109–178.

    MATH  Google Scholar 

  13. S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: spatial topological chaos. SIAM J. Appl. Math.,55 (1995), 1764–1781.

    Article  MATH  MathSciNet  Google Scholar 

  14. L.O. Chua and T. Roska, The CNN paradigm. IEEE Trans. Circuits Syst.,40 (1993), 147–156.

    MATH  MathSciNet  Google Scholar 

  15. L.O. Chua, CNN: A Paradigm for Complexity. World Scientific Series on Nonlinear Science, Series A31, 1998.

  16. L.O. Chua and L. Yang, Cellular neural networks: Theory. IEEE Trans. Circuit Syst.,35 (1988), 1257–1272.

    Article  MATH  MathSciNet  Google Scholar 

  17. L.O. Chua and L. Yang, Cellular neural networks: Applications. IEEE Trans. Circuit Syst.,35 (1988), 1273–1290.

    Article  MathSciNet  Google Scholar 

  18. C.-H. Hsu, Smale horseshoe of cellular neural networks. Int. J. Bifur. and Chaos,10, No. 12 (2000), 2119–2129.

    MATH  Google Scholar 

  19. C.-H. Hsu and S.-S. Lin, Existence and multiplicity of traveling waves in lattice dynamical system. J. Diff. Eqns.,164 (2000), 431–450.

    Article  MATH  MathSciNet  Google Scholar 

  20. C.-H. Hsu, S.-S. Lin and W. Shen, Traveling waves in cellular neural networks. Int. J. Bifur. and Chaos,9 (1999), 1307–1319.

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Juang and S.-S. Lin, Cellular neural networks: mosaic pattern and spatial chaos. SIAM J. Appl. Math.,60 (2000), 891–915.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Juang and S.-S. Lin, Cellular neural networks: defect pattern and spatial chaos. Preprint, 1998.

  23. P. Kevorkian, Snapshots of dynamical evolution of attractors from Chua’s oscillator. IEEE Trans. Circuit Syst.,40 (1993), 762–780.

    Article  MATH  Google Scholar 

  24. M.I. Malkin, On continuity of entropy of discontinuous mappings of the interval. Selecta Mathematica Sovietiea,8 (1989), 131–139.

    MATH  Google Scholar 

  25. J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems, II. IEEE Trans. Circuit Syst.,42 (1995), 852–856.

    MathSciNet  Google Scholar 

  26. de Melo and Van Strain, One-dimensional Dynamics. Spriger-Verlag, 1993.

  27. V.I. Nekorkin and L.O. Chua, Spatial disorder and wave fronts in a chain of coupled Chua’s circuits. Chua’s Circuit: A Paradigm for Chaos (ed. R.N. Madan), World Scientific Publishing Co. Pte. Ltd., 1993, 351–367.

  28. W. Parry, Intrinsic markov chains. Trans. Amer. Math. Soc.,112 (1964), 55–66.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. Orzo, Z. Vidnyanszky, J. Hamori and T. Roska, CNN model of the feature linked synchronized activities in the visual thalamo-cortical system. Proc. 1996 Fourth IEEE Int. Workshop on CNN and Their Applications, Seville, Spain, 24–26 June, 1996, 291–296.

  30. C. Robinson, Dynamical Systems. CRC Press, Boca Raton, FL, 1995.

    MATH  Google Scholar 

  31. P. Thiran, K.B. Crounse, L.O. Chua and M. Hasler, Pattern formation properties of autonomous cellular neural networks. IEEE Trans. Circuits Syst.,42 (1995), 757–774.

    Article  MathSciNet  Google Scholar 

  32. F. Werblin, T. Roska and L.O. Chua, The analogic cellular neural network as a bionic eye. Int. J. Circuit Theory Appl.,23 (1994), 541–569.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, National Central University, 320, Chung-Li, Taiwan

    Cheng-Hsiung Hsu

  2. Department of Applied Mathematics, National Chiao Tung University, 300, Hsinchu, Taiwan

    Song-Sun Lin

Authors
  1. Cheng-Hsiung Hsu
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  2. Song-Sun Lin
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Additional information

Work partially supported by NSC89-2155-M-008-012.

Work partially supported by NSC88-2115-M-009-001.

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Hsu, CH., Lin, SS. Spatial disorder of Cellular Neural Networks. Japan J. Indust. Appl. Math. 19, 143–161 (2002). https://doi.org/10.1007/BF03167451

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  • DOI: https://doi.org/10.1007/BF03167451

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