*Temper Coarse-Graining of ODE Systems using Parametrized Locally Invariant Manifolds*

**Likun Tan**

Carnegie Mellon University

Civil and Environmental Engineering

`likunt@andrew.cmu.edu`

**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.