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Carnegie Mellon Center for Nonlinear Analysis
Temper Coarse-Graining of ODE Systems using Parametrized Locally Invariant Manifolds


Likun Tan
Carnegie Mellon University
Civil and Environmental Engineering

Abstract: An approach for time averaging in nonlinear, autonomous ODE systems is demonstrated based on the idea of the Parametrized Locally Invariant Manifolds. By considering a fine scale physical theory characterized by an evolutionary system of equations and the running time average of the variables defined from the fine theory as a coarse representation of the fine scale phenomenon, one generates an augmented fine system with the introduction of the time-shifted fine variables and an evolutionary set of (more or less) closed equations for the coarse quantities. This coarse representation of dynamics could be of much larger time scale than the fine theory arising merely from the definition of time averaging. This method is illustrated on two model problems (macroscopic stress-strain behavior of an atomic chain based on the Frenkel-Kontorova model and time averaged response of a fine variable in 'Forced' Lorenz system with a user-constructed variable adding to the fine variable). Due to its efficiency in computation and flexibility in the establishment of coarse theory, this method has the potential on handing ODE systems with higher dimensions, such as molecular dynamics.