Georgia Institute of Technology College of Sciences

PDEs (such as Navier-Stokes) are in principle infinite-dimensional dynamical systems. However, recent studies support conjecture that the turbulent solutions of spatially extended dissipative systems evolve within an `inertial' manifold spanned by a finite number of 'entangled' modes, dynamically isolated from the residual set of isolated, transient degrees of freedom. We provide numerical evidence that this finite-dimensional manifold on which the long-time dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the `'physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors. (with Xiong Ding, H. Chate, E. Siminos and K. A. Takeuchi)