This group has been studying several global bifurcations for the Kuramoto-Sivashinsky equation (KSE) in detail. These bifurcations are triggered by the interactions between stable and unstable manifolds (one-dimensional and two-dimensional) of steady states and limit cycles. This approach is working to reduce the dimension of phase space by restricting flow to a three-dimensional approximate inertial manifold (AIM). Such a manifold can be described as providing a functional relation for the high (frequency) modes in terms of the low modes. There is strong numerical evidence to show that the local bifurcations for this reduced system are the same as that of the partial differential equations (PDE).
Michael S. Jolly, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota
David Norman, Graduate Student Researcher
Jaiok Roh, Graduate Student Researcher
99/162 |
"Perturbations of Normally Hyperbolic Manifolds with Applications to the Navier-Stokes Equations," V.A. Pliss and G.R. Sell, University of Minnesota Supercomputing Institute Research Report UMSI 99/162, October 1999. |
This group is seeking stronger evidence for the case of the global bifurcations they have observed. Their approach follows the theory of approximation dynamics. This theory requires the approximate inertial form to be a C1 perturbation of the PDE. If the C1 perturbation is small enough, certain hyperbolic structures are preserved under conditions quantifiable in terms of the linearized flow. However, to reduce the error for the AIM in some instances, one must increase the dimension of the manifold and lose the geometric understanding and low complexity of a three-dimensional phase space. In particular, by increasing the dimension of the phase space, one increases the dimension of the stable manifold making its computation unwieldy.
This group has implemented an algorithm developed to compute the inertial manifold to arbitrary C1-accuracy while keeping its dimension fixed. They have demonstrated convergence in a case where the exact manifold is known. With only a slight modification of the code, they successfully computed an inertial manifold with delay (IMd). This manifold enslaves the high modes at the present time in terms of the low modes at the present and the high modes at a small, fixed time delay in the past. Combining the IMd with multistep solvers for ordinary differential equations (ODEs), this group has demonstrated that they can compute a sensitive solution to the KSE as accurately in a three-dimensional phase space as with using many modes in a Galerkin approximation. Then, the strategy is to find the initial condition for the reduced space using the original algorithm for the inertial manifold and evolve the solution using a scheme based on the IMd.
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