Quadratic Dynamical Systems (Preliminary Version)

The main purpose of this paper is to promote the study of computational aspects, primarily the convergence rate, of nonlinear dynamical systems from a combinatorial perspective. We identify the class of symmetric quadratic systems. Such systems have been widely used to model phenomena in the natural sciences, and also provide an appropriate framework for the study of genetic algorithms in combinatorial optimisation. We prove several fundamental general properties of these systems, notably that every trajectory converges to a fixed point. We go on to give a detailed analysis of a quadratic system defined in a natural way on probability distributions over the set of matchings in a graph. In particular, we prove that convergence to the limit requires only polynomial time when the graph is a tree. This result demonstrates that such systems, though nonlinear, are amenable to quantitative analysis. 1 Introduction 1.1 Dynamical systems Many natural phenomena can be described by dynamical sy..