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Applications of Rosa's Sequence of Approximate Inertial Manifolds

Applications of Rosa's Sequence of Approximate Inertial Manifolds

A sequence of approximate inertial manifolds which converges to the exact manifold is implemented in several contexts. We first demonstrate the convergence in a case where the exact manifold is known. We then use the sequence to provide initial conditions for an accurate construction of global (un)stable manifolds. Lastly, we slightly alter the sequence to compute an inertial manifold with delay [Debuscche and Temam] and demonstrate that we can compute a sensitive solution to the Kuramoto-Sivashinsky equation just as accurately in a three-dimensional phase space as we can using a great many modes in a Galerkin approximation. The point in the last application is to not necessarily save in computational effort, but gain in geometric understanding.