The cover of this book shows ripples on a sandy beach and thus exemplifies, along with such things as the stripes of a zebra and the spots on a leopard's back, regular patterns arising in nature. This book introduces the wide range of mathematical methods used to analyze such patterns. As such, this text works as a springboard from a foundation in group theory and/or ODEs to applied modeling of pattern formation.
Basic supporting material on group theory and bifurcation theory follows an initial overview chapter. The reader should know that if the initial chapter seems elusive at parts, quite possibly explanation lies in the foundational material that follows.
A subject like this definitely benefits from visual presentation. This book includes 100 line diagrams, 30 half-tones, and eight color plates. At times this seems a little thin for the material. This is, of course, not a book in the class of Sir Theodore Andrea Cook’s logarithm picture album Curves of Life , but rather an upper-undergraduate textbook for mathematics students or an illustrated resource for readers in physics or biology.
The fusion of techniques from abstract and analytic fields makes for an engaging foray into understanding the mathematics of patterns and serves as a good source for active researchers in the field. While the book does not provide solutions to exercises, lecturers adopting the text in a course can access worked solutions online from the publisher. Topics covered include lattice patterns, envelope equations, traveling plane waves, spiral defect chaos, large-aspect-ratio systems, and the Cross-Newell equations.
1. What are natural patterns?; 2. A bit of bifurcation theory?; 3. A bit of group theory?; 4. Bifurcations with symmetry; 5. Simple lattice patterns; 6. Superlattices, hidden symmetries and other complications; 7. Spatial modulation and envelope equations; 8. Instabilities of stripes and travelling plane waves; 9. More instabilities of patterns; 10. Spirals, defects and spiral defect chaos; 11. Large-aspect-ratio systems and the Cross-Newell equations.