Time-averaged coarse variables for multi-scale dynamics
Marshall Slemrod,Amit Acharya +1 more
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In this paper, the authors define a strategy for producing a large class of "slow" variables (constants of fast motion) in an autonomous system of ODE without an a priori split into slow and fast components.Abstract:
Given an autonomous system of Ordinary Differential Equations without an a priori split into slow and fast components, we define a strategy for producing a large class of ‘slow’ variables (constants of fast motion) in a precise sense. The equation of evolution of any such slow variable is deduced. The strategy is to rewrite our system on an infinite dimensional “history” Hilbert space X and define our coarse observation as a functional on X.read more
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Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph
TL;DR: In this article, the authors present basic ideas, notions, and methods for reduction of reaction kinetics models: quasi-steady state, quasi-equilibrium, slow invariant manifolds, and limiting steps.
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Accelerated time integrator for multiple time scale homogenization
Robert Crouch,Caglar Oskay +1 more
TL;DR: An accelerated time domain homogenization methodology for prediction of material and structural failure under fatigue loading is presented in this article. But the method does not address the computational challenge associated with the scale disparity between the characteristic fatigue load period and the overall fatigue life.
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Modeling of slow time-scale behavior of fast molecular dynamic systems
TL;DR: Tan et al. as mentioned in this paper applied practical time averaging (PTA) to molecular dynamical (MD) systems, where a double-well interatomic potential among adjacent particles leads to a phase transition between two distinct equilibrium states.
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Coarse variables of autonomous ODE systems and their evolution
TL;DR: In this article, a coarse evolution model for ODEs is proposed for the deterministic, slow/coarse behavior of the ODE system, and a computational approximation is developed and tested on the Lorenz system and the Hald Hamiltonian system.
References
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Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
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Singularly perturbed ordinary differential equations with dynamic limits
Zvi Artstein,Alexander Vigodner +1 more
TL;DR: In this article, the qualitative limit behavior of the trajectories as the small parameter tends to zero is studied and invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach.
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Slow observables of singularly perturbed differential equations
TL;DR: In this article, singularly perturbed systems which may not possess a natural coordinate split into slow and fast dynamics are examined, and their limit behaviour is depicted as an invariant measure of the fast component drifted by the slow part of the system.
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On singularly perturbed ordinary differential equations with measure-valued limits
TL;DR: In this article, the limit behavior of solutions of a singularly perturbed system is examined in the case where the fast flow need not converge to a stationary point, and the topological convergence as well as information about the distribution of the values of the solutions can be determined in case that the support of the limit invariant measure is an asymptotically stable attractor.
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