Dynamics of spatially chaotic solutions of parabolic equations

We study parabolic systems with a potential non-linearity with one or many spatial variables. We describe a rather general and stable mechanism explaining the appearance and preservation of complicated stable spatial forms. The main idea consists in a description of the complexity of a solution in terms of its homotopy class. This class is a discrete-valued preserved quantity. The number of homotopy inequivalent solutions depends exponentially on the parameters of the equation. In our paper we discuss the connections between the dynamics of the solutions of parabolic systems with a complicated spatial structure and the properties of the Riemannian metric on the configuration space $\mathbb R^d$ generated by the Jacobian variational functional. The relationships between the lengths of the geodesics are reflected in the complexity of the spatial forms and in such dynamical properties as the attraction and repulsion of solitons.