Influence of noise and delay on reaction-diffusion recurrent neural networks

In this paper, the influence of the noise and delay upon the stability property of reaction-diffusion recurrent neural networks （RNNs） with the time-varying delay is discussed. The new and easily verifiable conditions to guarantee the mean value exponential stability of an equilibrium solution are derived. The rate of exponential convergence can be estimated by means of a simple computation based on these criteria.     

Random attractor for the Ladyzhenskaya model with additive noise

This paper studies the dynamical behavior of the Ladyzhenskaya model with additive noise. With some conditions, we prove that the generated random dynamical system has a compact random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space.     

On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.     

Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise

In this paper, we focus on the asymptotic behavior of the optimal filter where both signal and observation processes are driven by Lévy noises. Indeed, we study large deviations for the case where the signal-to-noise ratio is small by considering weak convergence arguments. To that end, we first prove the uniqueness of the solution of the controlled Zakai and Kushner–Stratonovich equations. For this, we employ a method which transforms the associated equations into SDEs in an appropriate Hilbert space. Next, taking into account the controlled analogue of Zakai and Kushner–Stratonovich equations, respectively, the large deviation principle follows by employing the existence, uniqueness and tightness of the solutions.     

An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

Intermittency for stochastic partial differential equations driven by strongly inhomogeneous space–time white noises

In this paper, the main topic is to investigate the intermittent property of the one-dimensional stochastic heat equation driven by an inhomogeneous Brownian sheet, which is a noise deduced from the study of the catalytic super-Brownian motion. Under some proper conditions on the catalytic measure of the inhomogeneous Brownian sheet, we show that the solution is weakly full intermittent based on the estimates of moments of the solution. In particular, it is proved that the second moment of the solution grows at the exponential rate. The novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.     

Small noise and long time phase diffusion in stochastic limit cycle oscillators

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor $\epsilon \searrow 0$ – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp ?\left(c{\epsilon }^{?2}\right)$, $c>0$, and we show both that on the time scale ${\epsilon }^{?2}$ the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.     

The stochastic wave equations driven by fractional and colored noises

We investigate a wave equation in the plane with an additive noise which is fractional in time and has a non-degenerate spatial covariance. The equation is shown to admit a process-valued solution. Also we give a continuity modulus of the solution, and the HSlder continuity is presented.     

Fully Data-Driven Time-Delay Interferometry with Time-Varying Delays

Raw space-based gravitational-wave data like laser interferometer space antenna's (LISA) phase measurements are dominated by laser frequency noise. The standard technique to make this data usable for gravitational-wave detection is time-delay interferometry (TDI), which cancels laser noise terms by forming suitable combinations of delayed measurements. To do so, TDI relies on inter-spacecraft distances and on how laser noise enters the interferometric data. The basic concepts of an alternative approach which does not rely on independent knowledge of temporal correlations in the dominant noise recently introduced. Instead, this automated principal component interferometry (aPCI) approach only assumes that one can produce some linear combinations of the temporally nearby regularly spaced phase measurements, which cancel the laser noise. Then the data is let to reveal those combinations, thus providing a set of laser-noise-free data channels. The authors' previous approach relies on the simplifying additional assumption that the filters which lead to the laser-noise-free data streams are time-independent. In LISA, however, these filters will vary as the constellation armlengths evolve. Here, a generalization of the basic aPCI concept compatible with data dominated by a still unmodeled but slowly varying dominant noise covariance is discussed. Despite its independence on any model, aPCI successfully mitigates laser frequency noise below the other noises' level, and its sensitivity to gravitational waves is the same as the state-of-the-art second-generation TDI, up to a 2% error.