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Mutual Synchronization, Control and Replication of Patterns and Waves in Coupled Lattices Composed of Bistable Units

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Synergetic Phenomena in Active Lattices

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

The content of this chapter is closely linked with the theme treated in Chap. 5. Here we also discuss active lattice systems composed of elements with bistable properties. However, the geometrical architecture of the systems to be used is more sophisticated. We consider here a multilayer lattice architecture, with interacting lattices or layers and hence a 3D geometry or anatomy. Many systems from various areas of science and technology have such multilayer 3D structure. Take, for instance Josephson superlattices consisting of many stacked tunnel junctions, artificial reaction-diffusion dynamical systems using molecular electronics technology, neural networks, layered porous media, Cellular Neural Networks and so on [6.15,6.1016,6.2224,6.26,6.27]. In spite of significant differences between all these cases we may expect some common hence universal properties associated with the multilayer geometry. The purpose of this chapter is to identify and describe such properties.

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References

  1. Babloyantz, A. and Lourenço, C., “Computation with chaos: A paradigm for cortical activity”, Proc. Natl. Acad. Sci. USA 91 (1994) 9027–9031.

    Article  ADS  Google Scholar 

  2. Binczak, S., Elibeck, J. C. and Scott, A. C., “Ephatic coupling of myelinated fibers”, Physica D 148 (2001) 159–179.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bose, A., “Symmetric and antisymmetric phases in parallel coupled nerve fibers”, SIAM J. Appl. Math. 55 (1995) 1650–1674.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brindley, J., Holden, A. V. and Palmer, A., A numerical model for reentry in weakly coupled parallel excitable fibres, in Nonlinear Wave Processes in Excitable Media, A. V. Holden, M. Markus and H. G. Othmer (Editors), (Plenum Press, New York, 1991), pp. 123–126.

    Google Scholar 

  5. Haken, H., Principles of Brain Functioning. A Synergetic Approach to Brain Activity, Behavior and Cognition (Springer-Verlag, Berlin, 1996).

    MATH  Google Scholar 

  6. Heagy, J. F., Carroll, T. L. and Pecora, L. M., “Synchronous chaos in coupled oscillator systems”, Phys Rev. E 50 (1994) 1874–1885.

    Article  ADS  Google Scholar 

  7. Heagy, J. F., Carroll, T. L. and Pecora, L. M., “Experimental and mumerical evidence for riddled basins in coupled chaotic systems”, Phys. Rev. Lett. 73 (1994) 3528–3531.

    Article  ADS  Google Scholar 

  8. Heagy, J. F., Carroll, T. L. and Pecora, L. M., “Desynchronization by periodic orbits”, Phys. Rev. E 52 (1995) 1253–1256.

    Article  ADS  Google Scholar 

  9. Heagy, J. F., Pecora, L. M. and Carroll, T. L., “Short wavelength bifurcations and size instabilities in coupled oscillator systems”, Phys. Rev. Lett. 74 (1995) 4185–4188.

    Article  ADS  Google Scholar 

  10. Hopfield, J. J., “Neural networks and physical systems with emergent collective computational abilities”, Proc. Natl. Acad. Sci. USA 79 (1982) 2554–2558.

    Article  MathSciNet  ADS  Google Scholar 

  11. Hopfield, J. J., “Pattern recognition computation using action potential timing for stimulus representation”, Nature 376 (1995) 33–36.

    Article  ADS  Google Scholar 

  12. Hoppensteadt, F. C. and Izhikevich, E. M., “Synchronization of laser oscillators, associative memory, and optical neurocomputing”, Phys. Rev. E 62 (2000) 4010–4013.

    Article  ADS  Google Scholar 

  13. Hoppensteadt, F. C. and Izhikevich, E. M., “Oscillatory neurocomputers with dynamic connectivity”, Phys. Rev. Lett. 82 (1999) 2983–2986.

    Article  ADS  Google Scholar 

  14. Keener, J. P., “Homogenization and propagation in the bistable equation”, Physica D 136 (2000) 1–17.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Kladko, K., Mitkov, I. and Bishop, A. R., “Universal scaling of wave propagation failure in arrays of coupled nonlinear cells”, Phys. Rev. Lett. 84 (2000) 4505–4508.

    Article  ADS  Google Scholar 

  16. Marquie, P., Comte, J. C. and Bilbault, J. M., “Contour detection using a two-dimensional diffusive nonlinear electrical network”, Proc. 2000 Int. Symposium On Nonlinear Theory and Its Applications (NOLTA 2000, Dresden, Germany, 2000), pp. 331–334.

    Google Scholar 

  17. Nekorkin, V. I., Kazantsev, V. B. and Velarde, M. G., “Image transfer in multilayered assemblies of lattices of bistable oscillators”, Phys. Rev. E 59 (1999) 4515–4522.

    Article  ADS  Google Scholar 

  18. Nekorkin, V. I., Kazantsev, V. B. and Velarde, M. G., “Mutual synchronization of two lattices of bistable elements”, Phys. Lett. A 236 (1997) 505–512.

    Article  ADS  Google Scholar 

  19. Nekorkin, V. I., Kazantsev, V. B., Artyukhin, D. V. and Velarde, M. G., “Wave propagation along interacting fiber-like lattices”, Eur. Phys. J. B 11 (1999) 677–685.

    Article  ADS  Google Scholar 

  20. Nekorkin, V. I., Kazantsev, V. B., Rabinovich, M. I. and Velarde, M. G., “Controlled disordered patterns and information transfer between coupled neural lattices with oscillatory states”, Phys. Rev. E 57 (1998) 3344–3351.

    Article  ADS  Google Scholar 

  21. Nekorkin, V. I., Kazantsev, V. B., Velarde, M. G. and Chua, L. O., “Pattern interaction and spiral waves in a two-layer system of excitable units”, Phys. Rev. E 58 (1998) 1764–1773.

    Article  ADS  Google Scholar 

  22. Palmer, A., Brindley, J. and Holden A. V., “Initiation and stability of reentry in two coupled excitable fibers”, Bull. Math. Biology 54 (1992) 1039–1056.

    MATH  Google Scholar 

  23. Panfilov, A. V. and Holden A. V., “Vortices in a system of two coupled excitable fibers”, Phys. Lett. A 147 (1990) 463–466.

    Article  ADS  Google Scholar 

  24. Pecora, L. M. and Carroll, T. L., “Synchronization of chaotic systems”, Phys. Rev. Lett. 64 (1990) 821–824.

    Article  MathSciNet  ADS  Google Scholar 

  25. Velarde, M. G., Nekorkin, V. I., Kazantsev, V. B. and Ross, J., “The emergence of form by replication”, Proc.Natl.Acad. Sci.USA 94 (1997) 5024–5027.

    Article  ADS  Google Scholar 

  26. Zinner, B., “Existence of traveling wavefront solutions for the discrete Nagumo equation”, SIAM J. Diff. Eqs. 96 (1992) 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  27. Zinner, B., “Stability of traveling wavefronts for the discrete Nagumo equation”, SIAM J. Math. Anal. 22 (1991) 1016–1020.

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Nekorkin, V.I., Velarde, M.G. (2002). Mutual Synchronization, Control and Replication of Patterns and Waves in Coupled Lattices Composed of Bistable Units. In: Synergetic Phenomena in Active Lattices. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56053-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-56053-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62725-5

  • Online ISBN: 978-3-642-56053-8

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