arXiv subjects

Eric Cooper

Publications and source records attributed to Eric Cooper.

3 recordsLinked to original sources

arXiv2019

Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.

math.DS↗

arXiv2017

Selection of quasi-stationary states in the Navier-Stokes equation on the torus

The two dimensional incompressible Navier-Stokes equation on $D_\delta := [0, 2\pi\delta] \times [0, 2\pi]$ with $\delta \approx 1$, periodic boundary conditions, and viscosity $0 < \nu \ll 1$ is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale $\mathcal{O}(e^{-\nu t})$ and have been shown to play a key role in its long-time evolution. Of particular interest is the role that $\delta$ plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will be selected if $\delta \neq 1$, while the dipole will be selected if $\delta = 1$. Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of $\delta$ controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all $\delta \approx 1$, there is an initial time period in which the high modes decay at the rapid rate $\mathcal{O}(e^{-t/\nu})$, while the low modes evolve at the slower $\mathcal{O}(e^{-\nu t})$ rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.

math.DS↗

arXiv2013

Multi-command Tactile Brain Computer Interface: A Feasibility Study

The study presented explores the extent to which tactile stimuli delivered to the ten digits of a BCI-naive subject can serve as a platform for a brain computer interface (BCI) that could be used in an interactive application such as robotic vehicle operation. The ten fingertips are used to evoke somatosensory brain responses, thus defining a tactile brain computer interface (tBCI). Experimental results on subjects performing online (real-time) tBCI, using stimuli with a moderately fast inter-stimulus-interval (ISI), provide a validation of the tBCI prototype, while the feasibility of the concept is illuminated through information-transfer rates obtained through the case study.

q-bio.NC↗