Pod systems: an equivariant ordinary differential equation approach to dynamical systems on a spatial domain

We present a class of systems of ordinary differential equations (ODEs), which we call 'pod systems', that offers a new perspective on dynamical systems defined on a spatial domain. Such systems are typically studied as partial differential equations, but pod systems bring the analytic techniques of ODE theory to bear on the problems, and are thus able to study behaviours and bifurcations that are not easily accessible to the standard methods. In particular, pod systems are specifically designed to study spatial dynamical systems that exhibit multi-modal solutions. A pod system is essentially a linear combination of parametrized functions in which the coefficients and parameters are variables whose dynamics are specified by a system of ODEs. That is, pod systems are concerned with the dynamics of functions of the form Ψ(s, t) = y1(t) φ(s; x1(t)) + ··· + yN(t) φ(s; xN(t)), where s in Rn is the spatial variable and φ: Rn × Rd → R. The parameters xi in Rd and coefficients yi in R are dynamic variables which evolve according to some system of ODEs, x-dot i = Gi(x, y) and y-dot i = Hi(x, y), for i = 1, ..., N. The dynamics of Ψ in function space can then be studied through the dynamics of the x and y in finite dimensions. A vital feature of pod systems is that the ODEs that specify the dynamics of the x and y variables are not arbitrary; restrictions on Gi and Hi are required to guarantee that the dynamics of Ψ in function space are well defined (that is, that trajectories are unique). One important restriction is symmetry in the ODEs which arises because Ψ is invariant under permutations of the indices of the (xi, yi) pairs. However, this is not the whole story, and the primary goal of this paper is to determine the necessary structure of the ODEs explicitly to guarantee that the dynamics of Ψ are well defined