Front propagation and mode-locking

in Coupled Map Lattices

Ricardo Carretero-González
Ph.D. Thesis
Department of Mathematical Sciences
Queen Mary and Westfield College

[ Download the whole thesis in PDF (4.3 MB) ]


0 Introduction

1 Extended states

    1.1 Homogeneous states and their stability
    1.2 Stability of homogeneous states in one-way CML
    1.3 Stability of homogeneous states in diffusive CML
    1.4 Stability of periodic global states

2 Localized states and propagating fronts

    2.1 Localized states
    2.2 Travelling velocity and basic properties

3 Travelling waves

    3.1 A bistable local map f
    3.2 Travelling waves in one-way CML
    3.3 Travelling waves in diffusive CML
    3.4 Mode-locking of the travelling velocity

4 A piece-wise linear local map

    4.1 The piece-wise linear local map
    4.2 The auxiliary map and its symbolic dynamics
          4.2.1 The auxiliary map for an increasing step state
          4.2.2 The auxiliary map for a decreasing step state
          4.2.3 Symbolic dynamics of minimal 1-states
    4.3 The travelling velocity
          4.3.1 Existence of the travelling velocity
          4.3.2 Continuity of the travelling velocity
          4.3.3 The travelling velocity is non-decreasing
          4.3.4 The velocity tree
    4.4 Mode-locking of the auxiliary map
          4.4.1 Mode-locking
          4.4.2 Stability diagram of the auxiliary map
          4.4.3 Velocity in the zero-gap case
    4.5 Representing the velocity tree with an integral lattice
          4.5.1 The integral lattice
          4.5.2 Concatenating sequences using the lattice representation
          4.5.3 Concatenating a family of sequences
          4.5.4 Unimodular transformations and envelopes

5 Higher order interface dynamics

    5.1 High order dynamics of minimal states
          5.1.1 Convergence to minimal states
          5.1.2 Two-dimensional auxiliary map
          5.1.3 The tongues for the two-dimensional case
          5.1.4 Higher order dynamics
    5.2 Mode-locking for smooth maps
          5.2.1 Mode-locking in one-way CMLs
          5.2.2 Mode-locking in diffusive CMLs
    5.3 Width of the mode-locking plateaus
          5.3.1 Mode-locking versus number of sites in the interface
          5.3.2 Mode-locking versus range of interaction

6 Conclusions and final remarks


You can mail comments to me, Ricardo Carretero,

Last updated: April 4th 1997

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