Coarse Variables of Autonomous ODE Systems and Their Evolution

TL;DR

This paper develops methods to model the slow, coarse behavior of autonomous ODE systems using two types of variables: time averages of phase functions and scalar functions that evolve autonomously, tested on Lorenz and Hamiltonian systems.

Contribution

It introduces practical approaches for constructing coarse evolution equations for autonomous ODE systems using invariant manifolds and autonomous scalar variables, with computational validation.

Findings

Coarse models accurately predict slow dynamics of Lorenz and Hamiltonian systems.

Invariant manifold approach effectively captures fast dynamics influence.

Autonomous scalar variables can be approximated to evolve independently.

Abstract

Given an autonomous system of ordinary differential equations (ODE), we consider developing practical models for the deterministic, slow/coarse behavior of the ODE system. Two types of coarse variables are considered. The first type consists of running finite time averages of phase functions. Approaches to construct the coarse evolution equation for this type are discussed and implemented on a 'Forced' Lorenz system and a singularly perturbed system whose fast flow does not necessarily converge to an equilibrium. We explore two strategies. In one, we compute (locally) invariant manifolds of the fast dynamics, parameterized by the slow variables. In the other, the choice of our coarse variables automatically guarantees them to be 'slow' in a precise sense. This allows their evolution to be phrased in terms of averaging utilizing limit measures (probability distributions) of the fast flow. Coarse evolution equations are constructed based on these approaches and tested against coarse response of the 'microscopic' models. The second type of coarse variables are defined as (non-trivial) scalar state functions that are required by design to evolve autonomously, to the extent possible, with the goal of being candidate state functions for unambiguously initializable coarse dynamics. The question motivates a mathematical restatement in terms of a first-order PDE. A computational approximation is developed and tested on the Lorenz system and the Hald Hamiltonian system.