Abstract
We prove the existence and we study the stability of the kinklike fixed points in a simple coupled map lattice (CML) for which the local dynamics has two stable fixed points. The condition for the existence allows us to define a critical value of the coupling parameter where a (multi) generalized saddle-node bifurcation occurs and destroys these solutions. An extension of the results to other CMLs in the same class is also displayed. Finally, we emphasize the property of spatial chaos for small coupling.
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References
J. D. Keeler and J. D. Farmer, Robust space-time intermittency and 1/f noise,Physica D 23:413–435 (1986).
A. Lambert and R. Lima, Stability of wavelengths and spacio-temporal intermittency,Physica D 71:390–411 (1994).
M. C. Cross and P. C. Hohenberg, Pattern formation outside equilibrium,Rev. Mod. Phys. 65:851–1113 (1993).
P. Collet and J. P. Eckmann,Instabilities and Fronts in Extended Systems (Princeton University Press, Princeton, New Jersey, 1990).
W. V. Saarloos, Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection,Phys. Rev. A 37:211–229 (1988).
B. Denardo, B. Galvin, A. Greenfield, A. Larraza, S. Putterman, and W. Wright, Observation of localized structures in nonlinear lattices: Domains walls and kinks,Phys. Rev. Lett. 68:1730–1733 (1992).
Y. S. Kivshar, Class of localized structures in nonlinear lattices,Phys. Rev. B 46:8652–8654 (1992).
J. P. Crutchfield and K. Kaneko, Phenomenology of spatio-temporal chaos,Direction in Chaos Hao Bai-Lin, ed. (World Scientific, Singapore, 1987), pp 272–353.
K. Kaneko, The coupled map lattices, inTheory and Applications of Coupled Map Lattices K. Kaneko, ed. (Wiley, New York, 1993).
B. Fernandez, Kink dynamics in one-dimensional coupled map lattice,Chaos 4(3):602–608 (1995).
S. Aubry, D. Escande, J. P. Gaspard, P. Manneville, and J. Villain,Structures et Instabilités (Les éditions de physique, Orsay, 1986).
V. S. Afraimovich and L. A. Bunimovich, Simplest structures in coupled map lattices and their stability,Random Computational Dynamics 1:423–444 (1993).
V. S. Afraimovich, L. Y. Glebsky, and V. I. Nekorkin, Stability of stationary states and topological spatial chaos in multidimensional lattice dynamical systems,Random Computational Dynamics 2:287–303 (1994).
R. S. Mackay and J.-A. Sepulchre, Multistability in networks of weakly coupled bistable units,Physica D 82:243–254 (1995).
V. S. Afraimovich and L. A. Bunimovich, Density of defects and spatial entropy in extended systems,Physica D 80:277–288 (1995).
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Fernandez, B. Existence and stability of steady fronts in bistable coupled map lattices. J Stat Phys 82, 931–950 (1996). https://doi.org/10.1007/BF02179796
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DOI: https://doi.org/10.1007/BF02179796