The Dynamics of Patterns

The following sections are included:

• Preface

• Contents

The diversity of patterns existing in the surrounding world usually arouses amazement not only in fiction novels but in scientific literature also. Perhaps still more curious is the fact that these diversified patterns are universal to a certain extent: hexagonal patterns on a giraffe's skin and hexagonal structures emerging at the onset of convection in a liquid layer; isolated patterns in colonies of microorganisms and soliton structures in hydrodynamics and in nonlinear optics; spiral chaos in heart muscle fibrillation and spiral turbulence in chemical systems; and a multitude of other examples.

Please refer to full text.

The following sections are included:

• Swift–Hohenberg equation

• Newell–Whitehead–Segel equation

• Coupled amplitude equations

• Phase equations

The following sections are included:

• The dissipative Ginzburg–Landau equation

• Nerve membrane excitation and the CGL equation

• Optical dynamics and the CGL equation

• Simple patterns in the CGL equation

• Phase equations revisited

• Gallery of phenomena

Regular patterns like that depicted in Fig. 1.2 that may generally be called ‘crystals’ are extremely simple. But the simplicity and regularity presents the following conundrum: Why does a nonlinear field evolve from an initially irregular state to a completely regular pattern? How does pattern selection occur in Nature? We have made some attempts to answer these questions in Chapter 2 when we drew attention to linear instabilities that behave selectively in space. It is these instabilities that usually gives rise to crystalline patterns which are so fascinating. We have observed, however, that these same instabilities may give birth to quite different patterns, even to disorder, in other regions of parameter space. Thus, the mere fact that the original instability is selective (|k| ≃ k₀) is insufficient for the formation of a spatially regular crystal picture. Of primary importance here is the next stage of the evolution: the interaction between steady regular and/or irregular mode structures. These processes of competition and selection are necessary for the birth of order from original disorder.

The following sections are included:

• Octagons, decagons, and dodecagons

• A generalized Swift-Hohenberg model

• The ‘turbulent’ crystal

The following sections are included:

• A simple model for domain walls

• Topological defects

• The birth of penta-hepta defects

• Dislocations and domain walls in Faraday ripples

The following sections are included:

• Bistable media

• Dynamical disorder of structures

• Particle interaction

• Chaotic scattering

The following sections are included:

• Active spirals
  • Spirals in the complex Ginzburg–Landau equation
  • Spirals in the FitzHugh-Nagumo model

• Passive spirals
  • Spirals in the Faraday experiment
  • Spirals in Rayleigh-Bénard convection

The following sections are included:

• Introduction

• Observations

• Models for vorticity generation
  • Marangoni wave model
  • The role of air

The following sections are included:

• Dictyostelium discoideum

• Esherichia coli

• Bacillus subtilis

The following sections are included:

• Introductory remarks

• Characteristics of space series

• The Grassberger-Procaccia algorithm

• Qualitative description of developing disorder

• Dynamical dimension of defect-mediated turbulence

The following sections are included:

• Introductory remarks

• Chaotic synchronization

• Coexistence of regular patterns and chaos

• Coarse grain spatio-temporal patterns

• Coherent patterns on a chaotic checkerboard

The following sections are included:

• Hallucinations

• Spatio-temporal patterns and information processing

The following sections are included:

• Appendix A: A Short Guide to Nonlinear Dynamics
  • Dynamical systems
    • Types of dynamical systems
    • Equilibrium states
    • Homoclinic and heteroclinic trajectories
    • Limit cycles
    • Quasiperiodic motion
  • Bifurcations
    • Bifurcations of equilibrium states
    • Birth of periodic motions
    • Change of stability in periodic motions
  • Chaotic Oscillations
    • Characteristics of chaos and the strange attractor
    • Chaotic Hamiltonian systems
    • Chaotic self-excited oscillations
  • Synchronization of oscillations
  • Dynamical chaos and turbulence

• Appendix B: Key Experiments in Pattern Formation
  • Parametrically excited patterns
    • Experiments with liquids
    • Experiments with granular material
  • Thermal convection
    • Rayleigh–Bénard convection
    • Patterns in Rayleigh–Bénard convection
    • Bénard–Marangoni convection
  • Diffusive chemical reactions
    • Turing patterns
    • Oscillating chemical reactions

• Bibliography

• Index