Ph.D. Thesis

Department of Mathematical Sciences

Queen Mary and Westfield College

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

** Index **

**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 **

** References **

*Last updated: April 4th 1997*

You are the visitor # since January 2002