zbMATH — the first resource for mathematics

Extended symbolic dynamics in bistable CML: Existence and stability of fronts. (English) Zbl 0933.37005
Summary: We consider a diffusive coupled map lattice (CML) for which the local map is piecewise affine and has two stable fixed points. By introducing a spatio-temporal coding, we prove the one-to-one correspondence between the set of global orbits and the set of admissible codes. This relationship is applied to the study of the (uniform) fronts’ dynamics. It is shown that, for any given velocity in $$[-1,1]$$, there is a parameter set for which the fronts with that velocity exist and their shape is unique. The dependence of the fronts’ velocity on the local map’s discontinuity is proved to be a devil’s staircase. Moreover, the linear stability of the global orbits which do not reach the discontinuity follows directly from our simple map. For the fronts, this statement is improved and as a consequence, the velocity of all the propagating interfaces is computed for any parameter. The fronts’ are shown to be also nonlinearly stable under some restrictions on the parameters. Actually, these restrictions follow from the co-existence of uniform fronts and non-uniformly travelling fronts for strong coupling. Finally, these results are extended to some $$C^\infty$$ local maps.

MSC:
 37B10 Symbolic dynamics 37B35 Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems 37E15 Combinatorial dynamics (types of periodic orbits) 82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
Full Text:
References:
 [1] Cross, M.C.; Hohenberg, P.C., Pattern formation outside equilibrium, Rev. mod. phys., 65, 851-1113, (1993) · Zbl 1371.37001 [2] Showalter, K., Quadratic and cubic reaction—diffusion fronts, Nonlinear science today, 4, 4, (1995) · Zbl 0839.92028 [3] Levine, H.; Reynolds, W.R., CML techniques for simulating interfacial phenomena in reaction—diffusion systems, Chaos, 2, 337-342, (1992) [4] Elkinani, J.; Villain, J., Growth roughness and instabilities due to the schwoebel effect: a one-dimensional model, J. phys. (France), 4, 949-973, (1994) [5] Aronson, D.G.; Weinberg, H.F., Multidimensional nonlinear diffusion arising in population genetics, Adv. math., 30, 33-76, (1978) · Zbl 0407.92014 [6] van Saarloos, W., Front propagation into unstable states: I marginal stability as a dynamical mechanism for velocity selection, Phys. rev. A, 37, 211-229, (1989) [7] Collet, P.; Eckmann, J.P., Instabilities and fronts in extended systems, (1990), Princeton University Press Princeton, NJ · Zbl 0732.35074 [8] Ben-Jacob, E.; Brand, H.; Dee, G.T.; Kramer, L.; Langer, J.S., Pattern propagation in nonlinear dissipative systems, Physica D, 14, 348-364, (1985) · Zbl 0622.76051 [9] Defontaines, A.-D.; Pomeau, Y.; Rostand, B., Chain of coupled bistable oscillators: a model, Physica D, 46, 210-216, (1990) · Zbl 0721.34032 [10] Erneux, T.; Nicolis, G., Propagating waves in discrete bistable reaction—diffusion systems, Physica D, 67, 237-244, (1993) · Zbl 0787.92010 [11] Keener, J.P., Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. appl. math., 47, 556-572, (1987) · Zbl 0649.34019 [12] Zinner, B.; Harris, G.; Hudson, W., Traveling wavefronts for the discrete Fisher’s equation, J. diff. eqs., 105, 46-62, (1993) · Zbl 0778.34006 [13] Afraimovich, V.S.; Nekorkin, V.I., Chaos of traveling waves in a discrete chain of diffusively coupled maps, Int. J. bif. chaos, 4, 631-637, (1994) · Zbl 0870.58049 [14] R. Carretero-González, D.K. Arrowsmith and F. Vivaldi, Mode-locking in CML, Physica D, submitted. [15] () [16] Kaneko, K., Chaotic travelling waves in a CML, Physica D, 68, 299-317, (1993) [17] Aubry, S.; Escande, D.; Gaspard, J.P.; Manneville, P.; Villain, J., Structures et instabilities, LES éditions de physique, (1986), Orsay [18] Fernandez, B., Existence and stability of steady fronts in bistable CML, J. statist. phys., 82, 931-950, (1996) · Zbl 1042.37534 [19] B. Fernandez and L. Raymond, The propagating fronts in bistable CML, J. Statist. Phys., to appear. · Zbl 0937.82030 [20] Bird, N.; Vivaldi, F., Periodic orbits of the sawtooth maps, Physica D, 30, 164-176, (1988) · Zbl 0648.58027 [21] R. Coutinho, Discontinuous rotations, preprint. [22] R. Coutinho and B. Fernandez, On the global orbits in a bistable CML, Chaos, to appear. · Zbl 0938.37052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.