Patterns of Dirt

Abstract

Many natural patterns appear when a simply structured equilibrium state is no longer preferred in comparison to a more complicated restructuring or rearrangement of the system. Our goal is typically a theoretical explanation or rationalization of the physical process, and invariably proceeds by way of mathematics; we formulate equations that describe the physical processes and seek to solve them in the appropriate context. Many standard techniques are available for the purpose to aid our analysis. For example, sometimes, hints about the patterns that will form can be extracted from a study of disturbances of infinitesimal amplitude, and so linear stability theory and decomposition into normal modes are our tools. Often, however, the ultimate, nonlinear mechanism of saturation is critical to selecting or shaping the forming pattern, and this cannot be revealed by linear stability analysis alone. Instead, we must advance into the nonlinear regime where we can use ideas from weakly nonlinear and dynamical systems theory complemented by numerical simulation.