Pattern Formation: An Introduction to MethodsFrom 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. |
Contents
A bit of bifurcation theory | 23 |
A bit of group theory | 52 |
Bifurcations with symmetry | 85 |
Simple lattice patterns | 134 |
Superlattices hidden symmetries and other complications | 168 |
Spatial modulation and envelope equations | 209 |
Instabilities of stripes and travelling plane waves | 243 |
More instabilities of patterns | 292 |
Spirals defects and spiral defect chaos | 325 |
Largeaspectratio systems and the CrossNewell equation | 380 |
| 408 | |
| 417 | |
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Common terms and phrases
absolutely irreducible amplitude equations bifurcation parameter bifurcation point bifurcation problem boundary conditions carrier wave centre manifold coefficients convection corresponding critical wavenumber d₁ defect defined derivatives dislocation domain drift Eckhaus instability eigenvalues equivariant example fixed point fixed-point subspace fluid free energy grain boundary group elements heteroclinic heteroclinic cycle hexagonal lattice hidden symmetries Hopf bifurcation irreducible representations irreps isotropy subgroup kB(k leading order linear linearising matrices modes modulated nonlinear terms nonzero one-dimensional origin pattern perturbations phase equation plane quasipatterns real constants reflection symmetry region roll solution rotation satisfy scaling shown in Figure solution branches space spatial spiral stable stationary point stationary solutions steady bifurcation stripes symmetry group theorem tion trajectory translation vector velocity wave wavelength wavenumber wavevector zigzag instability μΑ ат ӘА ди дф эт