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Pattern Formation An Introduction to Methods

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ISBN-10: 0521817501

ISBN-13: 9780521817509

Edition: 2005

Authors: Rebecca B. Hoyle

List price: $156.00
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From the stripes of a zebra and the spots on a leopard's back to the ripples on a sandy beach or desert dune, regular patterns arise everywhere in nature. The appearance and evolution of these phenomena have been a focus of recent research activity across several disciplines. This book provides an introduction to the range of mathematical theory and methods used to analyze and explain these often intricate and beautiful patterns. Bringing together several different approaches, from group theoretic methods to envelope equations and theory of patterns in large-aspect ratio-systems, the book also provides insight behind the selection of one pattern over another.
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Book details

List price: $156.00
Copyright year: 2005
Publisher: Cambridge University Press
Publication date: 3/17/2006
Binding: Hardcover
Pages: 434
Size: 8.31" wide x 10.12" long x 1.18" tall
Weight: 2.2
Language: English

What are natural patterns?
Reaction-diffusion systems
Faraday waves
Outline of the rest of the book
A bit of bifurcation theory
Flows, stationary points and periodic orbits
Local bifurcations from stationary points
Normal forms for bifurcations
Codimension-one bifurcations
A bit of group theory
Subgroups, quotient groups and conjugacy
Mappings of groups
Products of groups
Lie groups
Representations of groups
Isotypic decomposition
Bifurcations with symmetry
Ordinary differential equations with spatial symmetry
The equivariant branching lemma
Bifurcations in a box
Hopf bifurcations with symmetry
Heteroclinic cycles
Appendix: Proofs
Simple lattice patterns
Lattices and lattice patterns
Bifurcations on a lattice
Steady bifurcation on a square lattice
Steady bifurcation on a hexagonal lattice
Roll/stripe solutions
The Kuppers-Lortz instability
Hopf bifurcation on a one-dimensional lattice
Superlattices, hidden symmetries and other complications
Superlattice patterns
Mode interactions
Spatial-period-multiplying bifurcations
Pseudoscalar actions of E(2)
Hidden symmetries
Hidden symmetries and reflecting boundary conditions
Spatial modulation and envelope equations
Envelope equations for specific models
Envelope equations and symmetries
Free energies or Lyapunov functionals
Conservation of 'angular momentum'
Hopf bifurcations and the complex Ginzburg-Landau equation
Travelling waves and the nonlinear Schrodinger equation
Modulated hexagons
Instabilities of stripes and travelling plane waves
Universal instabilities of stripes
The Eckhaus instability
The zigzag instability
A general theory of phase dynamics
The cross-roll instability
Prandtl-number-dependent instabilities of convection rolls
The Benjamin-Feir instability
More instabilities of patterns
Instabilities of two-dimensional steady patterns
Drift instabilities
Galilean invariance and flat modes
Conservative systems and flat modes
Spirals, defects and spiral defect chaos
Types of isolated defect
Dislocation of a roll pattern
Amplitude grain boundaries
Domain boundaries between different patterns in systems with a free energy
Energetic considerations for rolls in finite domains
Spirals in oscillatory and excitable systems
Drifting and meandering spirals
Spiral defect chaos
Large-aspect-ratio systems and the Cross-Newell equation
Fully nonlinear patterns in large-aspect-ratio boxes
Stationary solutions of the Cross-Newell equation
Defect solutions of the Cross-Newell equation
Models with variational structure
Systems with mean drift