White Noise Perturbation Research Articles

ON THE STABILITY OF NON-FORCE-FREE MAGNETIC EQUILIBRIA IN STARS

The existence of stable magnetic configurations in white dwarfs, neutron stars and various non-convective stellar {regions} is now well recognized. It has recently been shown numerically that various families of equilibria, including axisymmetric mixed poloidal-toroidal configurations, are stable. Here we test the stability of an analytically-derived non force-free magnetic equilibrium, using three-dimensional magnetohydrodynamic simulations: the mixed configuration is compared with the dynamical evolution of its purely poloidal and purely toroidal components, both known to be unstable. The mixed equilibrium shows no sign of instability under white noise perturbations. {This configuration therefore provides a good description of magnetic equilibrium topology inside non-convective stellar objects and will be useful to initialize magneto-rotational transport in stellar evolution codes.

Passification Based Adaptive Control Under Coordinate-Parametric White Noise Disturbances

In the paper the basics of passification based adaptive control theory for stochastic continuous-time systems are introduced. Conditions for the mean square dissipativity of adaptive stabilization systems for a linear plant under coordinate-parametric white noise disturbances are obtained. A linear adaptive regulator with adaptation algorithm designed by the passification approach is proposed. The number of plant inputs may differ from that of outputs. The proof is based on the construction of a quadratic stochastic Lyapunov function. (In the case of purely parametric perturbations, the obtained conditions are known to be necessary and sufficient for the existence of a Lyapunov function with these properties.) Ultimate mean square boundedness (Levinson dissipativity) conditions for the designed closed loop system are obtained; it is shown that, in some special cases, the dissipativity of the closed loop system is preserved under white noise perturbations of arbitrary intensity. Stochastic G-passivity for nonsquare systems is introduced and necessary and sufficient conditions for strict stochastic G-passifiability are formulated.

Noisy heteroclinic networks

We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distribution in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion.

Fractional white noise perturbations of parabolic Volterra equations

Aim of this work is to extend the results of Clément, Da Prato and Prüss [Rend. Istit. Mat. Univ. Trieste 29: 207–220, 1997] on the fractional white noise perturbation with Hurst parameter H ∈ (0, 1). We will obtain similar results and it will turn out that the regularity of the solution u(t) increases with Hurst parameter H.

Effects of Time Delay and Noise on Asymptotic Stability in Human Quiet Standing Model

A human quiet standing stability is discussed in this paper. The model under consideration is proposed to be a delayed differential equation (DDE) with multiplicative white noise perturbation. The method of the center manifold is generalized to reduce a delayed differential equation to a two-dimensional ordinary differential equation, to study delay-induced instability or Hopf bifurcation. Then, the stochastic average method is employed to obtain the Itô equation. Thus, the top Lyapunov exponent is calculated and the necessary and sufficient condition of the asymptotic stability in views of probability one is obtained. The results show that the exponent is related to not only the strength of noise but also the delay, namely, the reaction speed of brain. The effect of the strength of noise on the human quiet standing losing stability is weak for a small delay. With the delay increasing, such effect becomes stronger and stronger. A small change in the strength of noise may destabilize the quiet standing for a large delay. It implies that a person with slow reaction is easy to lose the stability of his/her quiet standing.

Adaptive control of linear systems with coordinatewise-parametric perturbations of white noise type

Conditions for the mean-square dissipativity of adaptive stabilization systems for a linear object under coordinate-parametric perturbations of white noise type are obtained. A linear adaptive regulator with adjustable coefficients is chosen. For adjusting parameters, an adaptation algorithm is synthesized by the passification method. The number of inputs in objects under consideration may differ from that of outputs. The proof is based on the construction of a quadratic stochastic Lyapunov function. (In the case of purely parametric perturbations, the obtained conditions are known to be necessary and sufficient for the existence of a Lyapunov function with these properties.) Dissipativity conditions for the constructed closed system are obtained; it is shown that, in some special cases, the dissipativity of the closed system is preserved under white-noise perturbations of any intensity.

OPTIMAL ADAPTIVE PREDICTION FOR SISO SYSTEMS

An adaptive predictor that is optimal in the minimum mean-square error (MMSE) sense at each step is obtained for single-input, single-output (SISO) linear time-invariant discrete-time systems having general delay and white noise perturbation. The adaptive d-step-ahead predictor includes the uncertainty associated with the parameter and state estimates whereas conventional adaptive predictors, that are asymptotically optimal, ignore the uncertainty in the parameter estimates.

STABILITY OF STOCHASTIC DELAYED SIR MODEL

A stochastic version of the SIR model is investigated in this paper. The stability in probability of the steady state of the system is proved under suitable conditions on the white noise perturbations. Linearizations of the systems both with and without delay are given and their exponentially mean square stabilities are studied.

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamichysteresis

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes aGaussian character of the noise term describing interaction of the analyzed system with itscomplex surroundings. An additional assumption about the existence of timescaleseparation between the dynamics of the measured observable and the typical timescale ofthe noise allows external fluctuations to be modeled as temporally uncorrelated andtherefore white. However, in many natural phenomena the assumptions concerning theabove mentioned properties of ‘Gaussianity’ and ‘whiteness’ of the noise can beviolated. In this context, in contrast to the spatiotemporal coupling characterizinggeneral forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flightscorrespond to the class of Markov processes which can still be interpreted aswhite, but distributed according to a more general, infinitely divisible, stable andnon-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifestinteresting physical properties and have been addressed in various scenarios ofphysical transport exhibiting a superdiffusive behavior. Here we present a briefoverview of our recent investigations aimed at understanding features of stochasticdynamics under the influence of Lévy white noise perturbations. We find that thearchetypal phenomena of noise-induced ordering are robust and can be detected also insystems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinitevariance.

Firing pattern of bursting neurons under sinusoidal drive in mean-field modeling

Bursting has been observed in many sensory neurons, and is thought to be important in neural signaling, sleep, and some disorders of the brain. Bursting neurons have been studied via various types of conductance-based models at the single-neuron level. Important features of bursting have been reproduced by this type of model, but it is not certain how well the behavior of populations of bursting neurons can be represented solely by that of individual neurons. To study bursting neurons at the population level, a conductance-based model is incorporated into a mean-field model to yield a mean-field bursting model. The responses of the model to sinusoidal inputs are studied, showing that neurons with various different initial states are capable of phase-locked or intermittent firing, depending on their baseline voltage. Furthermore, depending on this voltage, the bursting frequency either slaves to the original unperturbed bursting frequency or approaches a steady value when the external driving frequency increases. Finally, use of white noise perturbations shows that the bursting frequency of the neurons remains the same even under a more general external stimulus.

Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.

Stability Analysis of Oscillation Modes in Quadruple-Push and Rucker's Oscillators

Oscillator systems composed by N sub-oscillators coupled through a symmetric linear network enable the combination of output power at the first or Nth harmonic component of the oscillation frequency of each sub-oscillator. However, they have the drawback of a possible coexistence of different oscillation modes, which limits their practical application. This paper presents an in-depth stability analysis of coexisting steady-state solutions in Rucker's oscillator and N -push oscillators. Criteria are provided to avoid undesired oscillation modes. The coupled system is described by means of a semianalytical formulation based on numerical models of the active subcircuits, extracted from harmonic-balance (HB) simulations. Each active subcircuit is composed by the transistor(s), feedback elements, and termination load. The use of the HB numerical models allows a realistic prediction of the behavior of the globally coupled oscillator. Alternatively, a graphical technique is provided to obtain the different oscillation modes using full HB simulations. The perturbation of the reduced-order nonlinear system enables the stability and phase-noise analysis of the steady-state oscillatory solutions. In the derived formulation for phase-noise analysis, both flicker- and white-noise perturbations are considered. The different techniques have been applied to a Rucker's oscillator at 3.9 GHz and a quadruple-push oscillator at 15.6 GHz.

Exponential growth and Gaussian—like fluctuations of solutions of stochastic differential equations with maximum functionals

In this paper we consider functional differential equations subjected to either instantaneous state-dependent noise, or to a white noise perturbation. The drift of the equations depend linearly on the current value and on the maximum of the solution. The functional term always provides positive feedback, while the instantaneous term can be mean-reverting or can exhibit positive feedback. We show in the white noise case that if the instantaneous term is mean reverting and dominates the history term, then solutions are recurrent, and upper bounds on the a.s. growth rate of the partial maxima of the solution can be found. When the instantaneous term is weaker, or is of positive feedback type, we determine necessary and sufficient conditions on the diffusion coefficient which ensure the exact exponential growth of solutions. An application of these results to an inefficient financial market populated by reference traders and speculators is given, in which the difference between the current instantaneous returns and maximum of the returns over the last few time units is used to determine trading strategies.

Nonlinear evolution of gravitational fragmentation regulated by magnetic fields and ambipolar diffusion

We present results from an extensive set of simulations of gravitational fragmentation in the presence of magnetic fields and ambipolar diffusion. The thin-sheet approximation is employed, with an ambient magnetic field that is oriented perpendicular to the plane of the sheet. Nonlinear development of fragmentation instability leads to substantial irregular structure and distributions of fragment spacings, fragment masses, shapes, and velocity patterns in model clouds. We study the effect of dimensionless free parameters that characterize the initial mass-to-flux ratio, neutral-ion coupling, and external pressure associated with the sheet. The average fragmentation spacing in the nonlinear phase of evolution is in excellent agreement with the prediction of linear perturbation theory. Both significantly subcritical and highly supercritical clouds have average fragmentation scales 〈 λ 〉 ≈ 2 π Z 0 , where Z 0 is the initial half-thickness of the sheet. In contrast, the qualitatively unique transcritical modes can have 〈 λ 〉 that is at least several times larger. Conversely, fragmentation dominated by external pressure can yield dense cluster formation with much smaller values of 〈 λ 〉 . The time scale for nonlinear growth and runaway collapse of the first core is ≈10 times the calculated growth time τ g,m of the eigenmode with minimum growth time, when starting from a uniform background state with small-amplitude white-noise perturbations. Subcritical and transcritical models typically evolve on a significantly longer time scale than the supercritical models. Infall motions in the nonlinear fully-developed contracting cores are subsonic on the core scale in subcritical and transcritical clouds, but are somewhat supersonic in supercritical clouds. Core mass distributions are sharply peaked with a steep decline to large masses, consistent with the existence of a preferred mass scale for each unique set of dimensionless free parameters. However, a sum total of results for various initial mass-to-flux ratios yields a broad distribution reminiscent of observed core mass distributions. Core shapes are mostly near-circular in the plane of the sheet for subcritical clouds, but become progressively more elongated for clouds with increasing initial mass-to-flux ratio. Field lines above the cloud midplane remain closest to vertical in the ambipolar-drift driven core formation in subcritical clouds, and there is increasing amount of magnetic field curvature for clouds of increasing mass-to-flux ratio. Based on our results, we conclude that fragmentation spacings, magnitude of infall motions, core shapes, and, especially, the curvature of magnetic field morphology, may serve as indirect observational means of determining a cloud’s ambient mass-to-flux ratio.

Spatial simulations of the Kelvin-Helmholtz instability in astrophysical jets

Aims. The long term magnetohydrodynamic stability of magnetized transonic/supersonic jets is numerically investigated using a spatial approach. We focus on two-dimensional linearly-unstable slab configurations where the jet is embedded in a flow-aligned uniform magnetic field of weak amplitude. We compare our results with previous studies using a temporal approach where longitudinally periodic domains were adopted. Methods. The finite-volume based versatile advection code is used to solve the full set of ideal compressible MHD equations. We follow the development of Kelvin-Helmholtz modes that are driven by a white noise perturbation continuously introduced at the jet inlet. Results. No noticeable difference is observed in spatial simulations versus analogous temporal ones during the linear and early non-linear evolution of the configuration. However, in the case of transonic flows, a different long-term scenario occurs in our spatial runs. Indeed, after the large-scale disruption of the flow, a sheath region of enhanced magnetic field encompassing the jet core forms along the whole flow. This provides a partial stabilization mechanism leading to enhanced stability for later times, which is almost independent of the initial magnitude of the magnetic field. The implication of this mechanism for the stability of astrophysical jets is discussed.

Stability of Ships with Water on Deck in Random Beam Waves

We study the stability of small ships with water on deck in random beam waves from a probability perspective. Previous research has shown that this kind of ship motion is governed by two dynamical regions: Homoclinic and heteroclinic, where the heteroclinic model emulates symmetric vessel capsize and the homoclinic model represents a vessel with an initial bias caused by water on deck. We investigate the stability and capsizing of ships in the homoclinic region using the probability method and nonlinear stochastic dynamics theory. Simplifying the random wave excitation to a periodic force and white noise perturbation, the random Melnikov mean square criterion is used to determine the parameter domain for the ship's stochastic chaotic motion. The probability density function of the roll response is calculated by solving the stochastic differential equations using the path integral method. A mathematical example is presented. It is found that in the chaotic parameter region, the probability density function of the system has two peaks. The response of the system will jump from one peak to another for large amplitudes of periodic excitation. This will lead to instability and even capsizing.

White-noise susceptibility and critical slowing in neurons near spiking threshold

We present mathematical and simulation analyses of the below-threshold noisy response of two biophysically motivated models for excitable membrane due to H. R. Wilson: a squid axon ("resonator") and a human cortical neuron ("integrator"). When stimulated with a low-intensity white noise superimposed on a dc control current, both membrane types generate voltage fluctuations that exhibit critical slowing down--that is, the voltage responsiveness to noisy input currents grows in amplitude while slowing in frequency--as the membrane approaches spiking threshold from below. We define threshold unambiguously as that dc current that renders a zero real eigenvalue for the Jacobian matrix for the integrator neuron, and, for the resonator neuron, as the dc current that gives a complex eigenvalue pair whose real part is zero. Using a linear Ornstein-Uhlenbeck analysis, we give exact small-noise expressions for the variance, power spectrum, and correlation function of the voltage fluctuations, and we derive the scaling laws for the divergence of susceptibility and correlation times for approach to threshold. We compare these predictions with numerical simulations of the nonlinear stochastic equations, and demonstrate that, provided the white-noise perturbations are kept sufficiently small, the linearized theory works well. These predictions should be testable in the laboratory using a current-clamped cell configuration. If confirmed, then the proximity of a neuron to its spike-transition point can be judged by measuring its subthreshold susceptibility to white-noise stimulation. We postulate that such temporally correlated fluctuations could provide a means of subthreshold signaling via gap-junction connections with neighboring neurons.

ON THE REPLICATOR DYNAMICS BEHAVIOR UNDER STRATONOVICH TYPE RANDOM PERTURBATIONS

This paper studies the behavior of replicator dynamics systems under White Gaussian Noise (WGN) perturbations in the Stratonovich form. Long-run behavior is described for the system with two pure strategies. The method of Lyapunov functions is used to find the sufficient conditions for asymptotic stability and strong instability of the pure strategies for the system with arbitrary number of pure strategies. Extinction of strictly dominated pure strategies is proven for any level of the noises.

Periodic-modulation-based blind channel identification for single-carrier block transmission with frequency-domain equalization

This paper proposes a periodic-modulation-based blind channel identification scheme for single-carrier (SC) block transmission with frequency-domain equalization (FDE). The proposed approach relies on the block system model and exploits the circulant channel matrix structure after the cyclic prefix is removed. It is shown that the set of linear equations relating the autocorrelation matrix of the block received signal and the product channel coefficients can be rearranged into one with a distinctive block circulant structure. The identification equations thus obtained lead to a very simple identifiability condition, as well as a natural formulation of the optimal modulating sequence design problem which, based on the block circulant signal structure, can be cast as a constrained quadratic problem that allows for a simple closed-form solution. The impact of the optimal modulating sequence on the peak-to-average power ratio (PAPR) is investigated. Also, it is shown that the optimal sequence results in a consistent channel estimate irrespective of white noise perturbation. Pairwise error probability analysis is used to examine the equalization performance and based on which several design tradeoffs are discussed. Simulation results are used for illustrating the performance of the proposed method.

Near-optimal controls of random-switching LQ problems with indefinite control weight costs

In this paper, we consider hybrid controls for a class of linear quadratic problems with white noise perturbation and Markov regime switching, where the regime switching is modeled by a continuous-time Markov chain with a large state space and the control weights are indefinite. The use of large state space enables us to take various factors of uncertain environment into consideration, yet it creates computational overhead and adds difficulties. Aiming at reduction of complexity, we demonstrate how to construct near-optimal controls. First, in the model, we introduce a small parameter to highlight the contrast of the weak and strong interactions and fast and slow motions. This results in a two-time-scale formulation. In view of the recent developments on LQ problems with indefinite control weights and two-time-scale Markov chains, we then establish the convergence of the system of Riccati equations associated with the hybrid LQ problem. Based on the optimal feedback control of the limit system obtained using the system of Riccati equations, we construct controls for the original problem and show that such controls are near-optimal. A numerical demonstration of a simple system is presented here.