Geology and geomorphology deal with some of the most striking patterns of Nature. From mountain ranges and mid-ocean ridges, to river networks and sand dunes, there is a whole family of forms, structures, and shapes that demand rationalization as well as mathematical description. In the various chapters of this volume, many of these patterns will be explored and discussed, and attempts will be made to both unravel the mathematical reasons for their very existence and to describe their dynamics in quantitative terms. In this introductory chapter, we discuss some of the methods that can be adopted in the study of patterns, and use the specific examples of convection - an evergreen classic in nonlinear fluid dynamics—and of the formation of aeolian ripples—another phenomenon that strikes the imagination of anybody who has been travelling in a sand desert.
Solitary Wave Rayleigh Number Hopf Bifurcation Dynamical System Theory Heteroclinic Orbit
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