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Coarse Variables of Autonomous ODE Systems and Their Evolution
Dynamical Systems · Submitted 15 Jun 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Coarse evolution equations are constructed based on these approaches and tested against coarse response of the 'microscopic' models.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given an autonomous system of ordinary differential equations (ODE), we consider developing practical models for the deterministic, slow/coarse behavior of the ODE system. | background 0.52 | 0.51 | 0.04 | hedged possibility 0.41 |
| s1 | Two types of coarse variables are considered. | contribution 0.81 | 0.52 | 0.02 | asserted 0.95 |
| s2 | The first type consists of running finite time averages of phase functions. | contribution 0.56 | 0.40 | 0.03 | asserted 0.98 |
| s3 | 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. | method 0.64 | 0.88 | 0.06 | asserted 0.99 |
| s4 | We explore two strategies. | method 0.85 | 0.78 | 0.02 | asserted 0.87 |
| s5 | In one, we compute (locally) invariant manifolds of the fast dynamics, parameterized by the slow variables. | method 0.99 | 0.83 | 0.03 | asserted 0.98 |
| s6 | In the other, the choice of our coarse variables automatically guarantees them to be 'slow' in a precise sense. | method 0.93 | 0.81 | 0.06 | asserted 0.99 |
| s7 | This allows their evolution to be phrased in terms of averaging utilizing limit measures (probability distributions) of the fast flow. | method 0.96 | 0.78 | 0.07 | asserted 0.91 |
| s8 | Coarse evolution equations are constructed based on these approaches and tested against coarse response of the 'microscopic' models. | method 0.62 | 0.94 | 0.12 | asserted 1.00 |
| s9 | 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. | contribution 0.89 | 0.81 | 0.06 | asserted 0.95 |
| s10 | The question motivates a mathematical restatement in terms of a first-order PDE. | method 0.64 | 0.76 | 0.04 | asserted 0.92 |
| s11 | A computational approximation is developed and tested on the Lorenz system and the Hald Hamiltonian system. | method 0.57 | 0.94 | 0.08 | asserted 1.00 |
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Original 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.
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What it cites · 3 sentences without a confident label
Uses the method or tool of 1 paper
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To numerically approximate the coarse dynamics ([ref]), we can apply a three-stage procedure suggested in [ref] : * Evolve the fast system ([ref]) with slow variables fixed at time step $s$ and $s+λ$, respectively to generate the Young measures as averages of $M$ Dirac masses at $M$ values of $f$, i.e. [equation] where $M$ is chosen to be large enough for the averages to converge. * Extrapolate to obtain the coarse variable at the next coarse step $s+T$ using the Forward Euler method, [equation] where $T$ is the coarse time step.
Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion · Introduction · 1.00
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For example, the averaging of the discrete KDV-Burgers type equation in [ref] required the explicit knowledge of the $N/2+1$ invariants of the fast flow with period $N$; such information, unfortunately, is generally not available for dynamical systems. * We have explored three options for constructing coarse evolution equations.
Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion · Introduction · 0.81
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