Abstract
In this chapter we consider lattice models with discrete time, i.e. dynamical systems with a discrete time, discrete space and continuous state and hence coupled map lattices (CML). CML describe the collective dynamics of a finite or infinite number of interacting finite-dimensional local dynamical systems (elements, units or cells) placed on the sites of a spatial lattice (the “physical” space). The dynamics of a single element is described by a point map. Thus, CML are characterized not only by their internal dynamics, i.e. that of the local elements, but also by the dynamics in the “physical” space. CML have been used to model a variety of phenomena including pattern formation, nonlinear waves, topological spatial chaos, distribution of defects in the spatial structures of nonlinear fields, reentry initiation in coupled parallel fibers, etc. [7.1–3, 7.6, 7.7, 7.9–16, 7.18, 7.20–22, 7.24, 7.25–31].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Afraimovich, V. S. and Bunimovich, L. A., “Simplest structures in coupled map lattices and their stability”, Int. J. Random Comput. Dyn. 1 (1993) 423–444.
Afraimovich, V. S. and Bunimovich, L. A., “Density of defects and spatial entropy in extended systems”, Physica D 80 (1995) 277–288.
Afraimovich, V. S. and Chow, S.-N., “Topological spatial chaos and homoclinic points of Zd-actions in lattice dynamical systems”, Jpn. J. Indust. Appl. Math. 12 (1995) 367–383.
Afraimovich, V. S., Glebsky, L. Yu. and Nekorkin, V. I., “Stability of stationary states and topological spatial chaos in multidimensional lattice dynamical systems”, Int. J. Random Comput. Dyn. 2 (1994) 287–303.
Afraimovich, V. S. and Nekorkin, V. I., “Chaos of travelling waves in a discrete chain of diffusively coupled maps”, Int. J. Bifurcation Chaos 4 (1994) 631–637.
Afraimovich, V. S. and Pesin, Ya., “Traveling waves in lattice models of multi-dimensional and multicomponent media: I. General hyperbolic properties”, Nonlinearity 6 (1993) 429–455.
Afraimovich, V. S. and Pesin, Ya., “Traveling waves in lattice models of multi-dimensional and multicomponent media: II. Ergodic properties and dimension”, Chaos 3 (1993) 233–241.
Arnéodo, A., Coullet, P. and Tresser, C., “Possible new strange attractors with spiral structure”, Commun. Math. Phys. 79 (1981) 573–579.
Ashwin, P., Buescu, J. and Stewart, I., “Bubbling of attractors and synchronization of chaotic oscillators”, Phys. Lett. A 193 (1994) 126–139.
Ashwin, P., Buescu, J. and Stewart, I., “From attractor to chaotic saddle: Tale of transverse instability”, Nonlinearity 9 (1996) 703–737.
Ashwin, P. and Terry, J., “On riddling and weak attractors”, Physica D 142 (2000) 87–100.
Bunimovich, L. A. and Sinai, Ya. G., Statistical Mechanics of Coupled Map Lattices, in Coupled Map Lattices: Theory and Applications, K. Kaneko (Editor), (Wiley, New York 1992).
Bunimovich, L. A., Livi, R., Martinez-Mekler, G. and Rutto, S., “Coupled trivial maps”, Chaos 2 (1992) 283–291.
Bunimovich, L. A. and Sinai, Ya. G., “Space-time chaos in coupled map lattices”, Nonlinearity 1 (1998) 491–516.
Carretero-Gonzalez, R., Arrowsmith, D. K. and Vivaldi, F., “Onedimensional dynamics for traveling fronts in coupled map lattices”, Phys. Rev. E 61 (2000) 1329–1336.
Chaté, H. and Courbage, M. (Editors), “Lattice dynamics”, Physica D 1–4 (1997).
Coutinho, R. and Fernandez, B., “On the global orbits in a bistable CML”, Chaos 7 (1997) 301–310.
Coutinho, R. and Fernandez, B., “Extended symbolic dynamics in bistable CML: Existence and stability of fronts”, Physica D 108 (1997) 60–80.
Feigenbaum, M. J., “Quantitative Universality for a Class of Nonlinear Transformations”, J. Stat. Phys. 19 (1978) 25–52.
Fernandez, B., “Kink dynamics in one-dimensional coupled map lattices”, Chaos 5 (1995) 602–608.
Fernandez, B., “Existence and stability of steady fronts in bistable CML”, J. Stat. Phys. 82 (1996) 931–950.
Glendinning, P., “Transitivity and blowout bifurcation in a class of globally coupled maps”, Phys. Lett. A 264 (1999) 303–310.
Haken, H., Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and Devices (Springer-Verlag, Berlin, 1983b).
Heagy, J. F., Carroll, T. L. and Pecora, L. M., “Desynchronization by periodic orbits”, Phys. Rev. E 52 (1995) 1253–1256.
Johnston, M. E., “Bifurcations of coupled bistable maps”, Phys. Lett. A 229 (1997) 156–164.
Kaneko, K., “Pattern dynamics in spatiotemporal chaos”, Physica D 34 (1989) 1–41.
Kaneko, K., “Spatiotemporal chaos in one-spatio and two-dimensional coupled map lattices”, Physica D 37 (1989) 60–82.
Kaneko, K., Simulating Physics with Coupled Map Lattices-Pattern Dynamics, Information Flow and Thermodynamics of the Spatio-temporal Chaos, in Formation, Dynamics and Statistics of Patterns, K. Kawasaki, A. Onuki and M. Suzuki (Editors), (World Scientific, Singapore, 1990), pp. 1–50.
Kaneko, K, “Overview of Coupled map lattices”, Chaos 2 (1992) 279–282.
Kaneko, K., “Chaotic traveling waves in a CML”, Physica D 68 (1993) 299–317.
Kaneko, K. and Tsuda, I., Complex Systems: Chaos and Beyond (Springer-Verlag, Berlin, 2001).
Nekorkin, V. I., “Spatial chaos in a discrete model of radiotechnical medium”, Radiotekh. Elektron. 37 (1992) 651–660 (in Russian).
Nekorkin, V. I., Kazantsev, V. B. and Velarde, M. G., “Synchronization in two-layer bistable coupled map lattices”, Physica D 151 (2001) 1–26.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nekorkin, V.I., Velarde, M.G. (2002). Spatio-Temporal Chaos in Bistable Coupled Map Lattices. In: Synergetic Phenomena in Active Lattices. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56053-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-56053-8_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62725-5
Online ISBN: 978-3-642-56053-8
eBook Packages: Springer Book Archive