dorsal/arxiv
View SchemaExtended Symbolic Dynamics in Bistable CML: Existence and Stability of Fronts
| Authors | R. Coutinho, B. Fernandez |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9609004 |
| URL | https://arxiv.org/abs/patt-sol/9609004 |
| DOI | 10.1016/S0167-2789(97)82005-2 |
Abstract
We consider a diffusive Coupled Map Lattice (CML) for which the local map is piece-wise 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 velocity of the fronts 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.
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"abstract": "We consider a diffusive Coupled Map Lattice (CML) for which the local map is\npiece-wise affine and has two stable fixed points. By introducing a\nspatio-temporal coding, we prove the one-to-one correspondence between the set\nof global orbits and the set of admissible codes. This relationship is applied\nto the study of the (uniform) fronts\u0027 dynamics. It is shown that, for any given\nvelocity in $[-1,1]$, there is a parameter set for which the fronts with that\nvelocity exist and their shape is unique. The dependence of the velocity of the\nfronts on the local map\u0027s discontinuity is proved to be a Devil\u0027s staircase.\nMoreover, the linear stability of the global orbits which do not reach the\ndiscontinuity follows directly from our simple map. For the fronts, this\nstatement is improved and as a consequence, the velocity of all the propagating\ninterfaces is computed for any parameter. The fronts are shown to be also\nnonlinearly stable under some restrictions on the parameters. Actually, these\nrestrictions follow from the co-existence of uniform fronts and non-uniformly\ntravelling fronts for strong coupling. Finally, these results are extended to\nsome $C^{\\infty}$ local maps.",
"arxiv_id": "patt-sol/9609004",
"authors": [
"R. Coutinho",
"B. Fernandez"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1016/S0167-2789(97)82005-2",
"title": "Extended Symbolic Dynamics in Bistable CML: Existence and Stability of Fronts",
"url": "https://arxiv.org/abs/patt-sol/9609004"
},
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