Invariant manifolds for dissipative systems

A method is given for a study of a nonlinear evolution equation for finding “slow” invariant manifolds. The method is studied for the evolution problem u̇=−Au + Au ,u / u ,u u, u 0 =u0, where A is a linear, self-adjoint, possibly unbounded operator in a Hilbert space. Global existence and uniqueness of the solution to this problem are proven. Asymptotic behavior of the solution as t→ is studied. Analytic solution of the above nonlinear evolution problem is found. Conditions are given on the spectrum of A and on the initial data u0 for the trajectory u t not to have a strong limit in H as t→ and not to stay in any finite-dimensional space. In this sense the motion is chaotic. © 2009 American Institute of Physics. DOI: 10.1063/1.3105924