2,796 research outputs found
Sharp estimates for the number of degrees of freedom for the damped-driven 2D Navier--Stokes equations
We derive upper bounds for the number of asymptotic degrees (determining
modes and nodes) of freedom for the two-dimensional Navier--Stokes system and
Navier-Stokes system with damping. In the first case we obtain the previously
known estimates in an explicit form, which are larger than the fractal
dimension of the global attractor. However, for the Navier--Stokes system with
damping our estimates for the number of the determining modes and nodes are
comparable to the sharp estimates for the fractal dimension of the global
attractor. Our investigation of the damped-driven 2D Navier--Stokes system is
inspired by the Stommel--Charney barotropic model of ocean circulation where
the damping represents the Rayleigh friction. We remark that our results
equally apply to the Stommel--Charney model
Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers
In this paper we introduce a finite-parameters feedback control algorithm for
stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped
nonlinear wave equations and the nonlinear wave equation with nonlinear damping
term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation.
This algorithm capitalizes on the fact that such infinite-dimensional
dissipative dynamical systems posses finite-dimensional long-time behavior
which is represented by, for instance, the finitely many determining parameters
of their long-time dynamics, such as determining Fourier modes, determining
volume elements, determining nodes , etc..The algorithm utilizes these finite
parameters in the form of feedback control to stabilize the relevant solutions.
For the sake of clarity, and in order to fix ideas, we focus in this work on
the case of low Fourier modes feedback controller, however, our results and
tools are equally valid for using other feedback controllers employing other
spatial coarse mesh interpolants
Importance-sampling computation of statistical properties of coupled oscillators
We introduce and implement an importance-sampling Monte Carlo algorithm to
study systems of globally-coupled oscillators. Our computational method
efficiently obtains estimates of the tails of the distribution of various
measures of dynamical trajectories corresponding to states occurring with
(exponentially) small probabilities. We demonstrate the general validity of our
results by applying the method to two contrasting cases: the driven-dissipative
Kuramoto model, a paradigm in the study of spontaneous synchronization; and the
conservative Hamiltonian mean-field model, a prototypical system of long-range
interactions. We present results for the distribution of the finite-time
Lyapunov exponent and a time-averaged order parameter. Among other features,
our results show most notably that the distributions exhibit a vanishing
standard deviation but a skewness that is increasing in magnitude with the
number of oscillators, implying that non-trivial asymmetries and states
yielding rare/atypical values of the observables persist even for a large
number of oscillators.Comment: 11 pages, 4 figures; v2: minor changes, close to the published
version, title changed to conform to PRE guideline
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
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