Georgia Institute of Technology College of Sciences

Theoretical aspects: If a smooth dynamical system on a compact invariant set is structurally stable, then it has the shadowing property, that is, any pseudo (or approximate) orbit has a true orbit nearby. In fact, the system has the Lipschitz shadowing property, that is, the distance between the pseudo and true orbit is at most a constant multiple of the local error in the pseudo orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement for discrete dynamical systems, that is, if a discrete dynamical system has the Lipschitz shadowing property, then it is structurally stable. In this talk this result will be reviewed and the analogous result for flows, obtained jointly with S. Pilyugin and S. Tikhomirov, will be described. Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak. A rigorous numerical method for establishing the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous system of ordinary differential equations is presented. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently ``good''. It turns out that inversion of the operator is equivalent to the solution of a boundary value problem for a nonautonomous inhomogeneous linear difference equation. A numerical procedure is given to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse (the latter determines how ``good'' the approximating orbit must be).

Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on each vertex, and directed edges represent couplings among vertex systems. Questions such as stability in the large, synchronization, and complexity in terms of dynamic clusters are of interest. A more recent approach is to investigate the connections between network topology and dynamical behaviours. I will present some recent results on the construction of global Lyapunov functions for coupled systems on networks using a graph theoretic approach, and show how such a construction can help us to establish global behaviours of compelx models.

We will explain the concept of aproximate well-supported Nash equilibrium and show that one must consider equilibria with large supports to achieve good approximation ratio. Our arguments use tools from probabilistic, extremal and additive combinatorics. Joint work with Y. Anbalagan, R. Savani and A. Vetta.

We say that a cover of surfaces S-> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.