Spiral waves in excitable media due to noise and periodic forcing

We investigate the jointly driven effects of external periodic forcing and Gaussian white noise on meandering spiral waves in excitable media with FitzHugh-Nagumo local dynamics. Interesting phenomena resulted from various forcing periods are found, for example, piece-wise line drift, intermittent straight-line drift and so on. We also observe new type of breakup of spiral wave between entrainment bands with 1:1 and 2:1. It is believed that the occurrence of the new type is relevant to the appearance of local bidirectional propagation window. There exist optimized noise intensities which can induce the broadest entrainments and Arnold tongues. Such a phenomenon is referred to as stochastic resonance. It is also observed that the noise makes significant effects on the spiral wave with straight-line drift. Via the tip Fourier spectrum, the varying of tip motion with external periods on the resonance band is interpreted.     

Type I vs. type II excitable systems with delayed coupling

Excitable and oscillatory dynamics of delayed locally coupled type I and type II excitable systems is analyzed. Diffusive and sigmoid coupling have been considered. It is shown that the stability and the patterns of exactly synchronous oscillations depend on the type of excitability and the type of coupling. However, within the same class, characterized by the excitability type and the coupling the dynamics qualitatively depends only on whether the number of units is even or odd.     

Robust optimization for performance tuning of modern database systems

Issues regarding design and management of database systems have been studied by applying operations research (OR) techniques. The purpose of this study is to propose a new alternative towards database performance tuning for query-processing needs of modern database systems from the perspective of operations research using robust optimization. We use a query-driven approach to specify database structures (schema) so that they are robust to uncertainty and dynamics of queries in a changing environment and allow fast and timely information retrieval and exchange. Instead of applying hardware tuning or traditional database tuning techniques, we examine queries by their types and properties to derive database structures that are robust at efficiently processing future queries of any type. This query-driven approach improves the efficiency of processing queries by setting up database structures based on the queries’ information needs. This new methodology provides a new approach of tuning database performance that is robust to unexpected changes and dynamics. To further demonstrate the idea, we develop a robust optimization model using a non-linear von Neumann–Morgenstern expected utility function and present two computational examples.     

Integro-differential systems with fuzzy noise

For a controlled integro-differential equation with fuzzy noise, we introduce the notions of a fuzzy bundle of trajectories and a fuzzy reachability set and prove some properties of fuzzy bundles. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1322–1330, October, 2007.     

Statistical structuring theory in parametrically excitable dynamical systems with a Gaussian pump

Based on the idea of the statistical topography, we analyze the problem of emergence of stochastic structure formation in linear and quasilinear problems described by first-order partial differential equations. The appearance of a parametric excitation on the background of a Gaussian pump is a specific feature of these problems. We obtain equations for the probability density of the solutions of these equations, whence it follows that the stochastic structure formation emerges with probability one, i.e., for almost every realization of the random parameters of the medium.     

Control of synchronization and spiking regularity by heterogenous aperiodic high-frequency signal in coupled excitable systems

This paper investigates the synchronization and spiking regularity induced by heterogenous aperiodic (HA) signal in coupled excitable FitzHugh–Nagumo systems. We found new nontrivial effects of couplings and HA signals on the firing regularity and synchronization in coupled excitable systems without a periodic external driving. The phenomenon is similar to array enhanced coherence resonance (AECR), and it is shown that AECR-type behavior is not limited to systems driven by noises. It implies that the HA signal may be beneficial for the brain function, which is similar to the role of noise. Furthermore, it is also found that the mean frequencies, the amplitudes and the heterogeneity of HA stimuli can serve as control parameters in modulating spiking regularity and synchronization in coupled excitable systems. These results may be significant for the control of the synchronized firing of the brain in neural diseases like epilepsy.     

Representation of systems disturbed by wide band noise

In this paper, a method of handling and working with wide band noise is developed. We represent wide band noise as a distributed delay of white noise and use it to reduce a nonlinear system disturbed by wide band noise to a nonlinear system disturbed by white noise. An application of this reduction to a nonlinear filtering problem under a wide band noise disturbance is discussed.     

Inversion of linear dynamical systems in the presence of additive noise

We consider the problem of inversion of a stationary dynamical system (i.e., the generation of an estimate of the unknown output signal on the basis of the known output signal) in the presence of an unknown perturbation acting “not in the same channel” as the input to be estimated. We present necessary and sufficient conditions for the inversion of such systems and suggest inversion algorithms.     

Effect of noise on the pattern formation in an epidemic model

We present novel numerical evidence of complicated phenomenon controlled by noise in a spatial epidemic model. The number of the spot is decreased as the noise intensity being increased, which we show by performing a series of numerical simulations. Moreover, when the noise intensity and temporal correlation are both large enough, the model dynamics exhibits a noise controlled transition from spotted pattern to stripe growth. In addition to that, we show in details the number of the spotted and stripe pattern, with the identification of a wide range of noise intensity and temporal correlation. The obtained results show that noise plays an important role in the pattern formation of the epidemic model, which may provide guidance to prevent and control the spread of disease. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Dynamics of Kolmogorov systems of competitive type under the telegraph noise

This paper studies the dynamics of Kolmogorov systems of competitive type under the telegraph noise. The telegraph noise switches at random two Kolmogorov competition-type deterministic models. The aim of this work is to describe the omega-limit set of the system and investigates properties of stationary density.     

Stability,coherent spiking and synchronization in noisy excitable systems with coupling and internal delays

We study the onset and the adjustment of different oscillatory modes in a system of excitable units subjected to two forms of noise and delays cast as external or internal according to whether they are associated with inter- or intra-unit activity. Conditions for stability of a single unit are derived in case of the linearized perturbed system, whereas the interplay of noise and internal delay in shaping the oscillatory motion is analyzed by the method of statistical linearization. It is demonstrated that the internal delay, as well as its coaction with external noise, drive the unit away from the bifurcation controlled by the excitability parameter. For the pair of interacting units, it is shown that the external/internal character of noise primarily influences frequency synchronization and the competition between the noise-induced and delay-driven oscillatory modes, while coherence of firing and phase synchronization substantially depend on internal delay. Some of the important effects include: (i) loss of frequency synchronization under external noise; (ii) existence of characteristic regimes of entrainment, where under variation of coupling delay, the optimized unit (noise intensity fixed at resonant value) may be controlled by the adjustable unit (variable noise) and vice versa, or both units may become adjusted to coupling delay; (iii) phase synchronization achieved both for noise-induced and delay-driven modes.     

Nondegenerate systems and generic properties of the integrable Hamiltonian systems

The goal of the paper is to clarify whether the nondegenerate Hamiltinian systems are “typical” among all integrable systems. The importance of this problem is emphasized by a theorem of Fomenko-Zieschang, in which an isoenergy invariant determining the nondegenerate systems up to topological equivalence is constructed. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 184–192.     

Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing

A thorough evaluation of stochastic resonance with tuning system parameters in bistable systems is presented as a nonlinear signal processor. It is shown that the output signal-to-noise ratio obtained by adjusting systems parameters can exceed that by tuning noise intensity, especially when the input noise intensity is already beyond the resonance region. It is demonstrated that the theory and the method presented here can markedly improve the output signal-to-noise ratio, and minimize phase lag as well as the distortion of the system output signal with multi-frequency.     

Malliavin calculus for infinite-dimensional systems with additive noise

We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the dynamics, we develop in this setting a partial counterpart of Hörmander's classical theory of Hypoelliptic operators. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the Lebesgue measure. We also apply our results to concrete SPDEs such as a Stochastic Reaction Diffusion Equation and the Stochastic 2D Navier-Stokes System.     

Ergodicity in infinite Hamiltonian systems with conservative noise

Summary. We study the stationary measures of an infinite Hamiltonian system of interacting particles in ℝ ³ subject to a stochastic local perturbation conserving energy and momentum. We prove that the translation invariant measures that are stationary for the deterministic Hamiltonian dynamics, reversible for the stochastic dynamics, and with finite entropy density, are convex combination of “Gibbs” states. This result implies hydrodynamic behavior for the systems under consideration. Received: 17 December 1994/In revised form: 12 April 1996     

On a classification of dynamic systems subject to noise

A classification of alternative mathematical schemes which determine possible impositions of noise on dynamic systems either of a continuous or a discrete formulation in time is presented. We refer to the results obtained concerning the influence of noise on the correlation dimension and on the largest Lyapunov exponent. A method of constructing models for effecting predictions of time series with a finite correlation dimension is presented. Numerical results concerning the influence of various level of additive noise to the Henon attractor are also provided.     

A-stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.This author was partially supported by the Italian Consiglio Nazionale delle Ricerche.     

Some algebraic properties of the hyperbolic systems

Abstract The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6]. In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Such a property is closely connected with some interesting inequalities on the eigenvalues of a hyperbolic matrix. We expect that this technique applies also to other problems. Keywords: First order hyperbolic systems, Quasi-symmetrizer, Glaeser inequality