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Center Manifold Theory and Computation Using a Forward Backward Approach
Schaal, Emily E
Schaal, Emily E
Abstract
The center manifold, an object from the field of differential equations, is useful in describing the long time behavior of the system. The most common way of computing the center manifold is by using a Taylor approximation. A different approach is to use iterative methods, as presented in Fuming and Kupper, 1994, Dellnitz and Hohmann, 1997, and Jolly and Rosa, 2005. In particular, Jolly and Rosa present a method based on a discretization of the Lyapunov-Perron (L-P) operator. One drawback is that this discretization can be expensive to compute and have error terms that are difficult to control. Using a similar framework to Jolly and Rosa,, we develop a forward-backward integration algorithm based on a boundary value problem derived from the operator. We include the details of the proofs that support this formulation; notably, we show that the operator is a contraction mapping with a fixed point that is a solution to the differential equation in our function space. We also show the first step in the induction to prove the existence of a $\mathcal{C}^k$ center manifold. We demonstrate the algorithm with Runge-Kutta (R-K) methods of $\mathcal{O}(k)$ and $\mathcal{O}(k^2)$. Finally, we present an application of our algorithm to studying a semilinear elliptic boundary value problem from Kirchgassner, 1982.
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2017-04-01
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Mathematics