Number of effective degrees of freedom of infinite dimensional systems

In ubiquitous natural and laboratory situations dissipative nonlinear partial differential equations (PDEs) are used to model the phenomena of pattern formation, space-time chaos and turbulence etc. Despite their infinite dimensional nature it was believed that albeit some trivial transient decaying process the relevant dynamics of these PDEs occurs on a finite dimensional manifold, named the inertial manifold (IM). The concept of IM thus opens up the possibility to model an infinite dimensional PDE system by a finite dimensional one. Although conceptually important the merit of IM is largely unexplored limited by the complexity of the necessary studying tools.

By using the method of Lyapunov analysis a hyperbolic separation between two sets of Lyapunov modes was detected and the finite number of mutually entangled physical modes were conjectured being associated with the effective dynamics on the inertial manifold (IM). A projection method is proposed which allows us to show the first direct relation between physical modes and the phase space structure of IM in support of the mentioned conjecture.