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.
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