A Variational Principle for Computing Slow Invariant Manifolds in Dissipative Dynamical Systems

Authors:
Dirk Lebiedz;Jochen Siehr;Jonas Unger
Affiliations:
dirk.lebiedz@biologie.uni-freiburg.de and jonas.unger@zbsa.de;jochen.siehr@iwr.uni-heidelberg.de;-
Venue:
SIAM Journal on Scientific Computing
Year:
2011

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Abstract

A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for finite-dimensional dynamical systems using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically exactly identifies the slow invariant manifold in the limit of either an infinite time horizon of the variational problem with fixed spectral gap or infinite spectral gap with a fixed finite time horizon. Numerical results are presented for the linear and nonlinear model problems as well as for a more realistic higher-dimensional chemical reaction mechanism.