A combinatorial/topological approach to nonlinear dynamics with applications to regulatory network dynamics

Konstantin Mischaikow
Department of Mathematics, Rutgers, The State University of New Jersey

The classical theory of nonlinear dynamical system focuses on the existence and structure of sets of trajectories. Bifurcation theory makes clear that these objects can change dramatically with small changes in the nonlinearities. Motivated by questions from systems biology, where neither parameters nor nonlinearities are known with any reasonable level of precision, I will discuss a novel approach to dynamics based on a combinatorial representation of the dynamics and the use of algebraic topological invariants to identify lower bounds on the structure. To provide a concrete realization of these techniques I describe work aimed at identifying simple models for the restriction point dynamics in mammalian cells.