Connecting orbits describe the paths along which changes occur in deterministic systems. In fact, every such dynamical system can be decomposed into so-called recurrent components and connecting orbits between them. This is most pronounced in gradient systems, where this dichotomy between equilibria and connections leads to the definition of the Morse-Floer complex. While equilibria are relatively manageable objects analytically, one usually has to resort to numerical simulations for studying connecting orbits. And while numerical calculations offer clear pictures of the dynamics, the information the numerics provide is non-rigorous. Strikingly, today's advances in computer speed and algorithm development make it possible to utilize the power and robustness of topological-analytic methods to rigorously validate computational results. Indeed, the past decade has seen enormous advances in the development of computer assisted proofs in dynamics. In this talks we will discuss recent progress in the rigorous computation of connecting orbits, and directions for further research will be outlined.

In this talk I will present formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. I will also analyze the limit cases of harmonic map heat flow and the Schrödinger map flow. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic. This joint work with Jan Bouwe van den Berg.

We investigate the formation and movement of self-organizing collectives of individuals in homogeneous environments. We review a hyperbolic system of conservation laws based on the assumption that the interactions governing movement depend not only on distance between individuals, but also on whether neighbours move towards or away from the reference individual. The inclusion of direction-dependent communication mechanisms significantly enriches the model behavior; the model exhibits classical patterns such as stationary pulses and traveling trains, but also novel patterns such as zigzag pulses, breathers, and feathers. The same enrichment of model behavior is observed when we include direction-dependent communication mechanisms in individual-based models.