Noise induced patterning in periodic systems with conserved dynamics

We study pattern formation in periodic systems with conserved dynamics driven by both thermally sustained flux and external (athermal) flux. Assuming the stochastic nature of each kind of flux we discuss noise induced patterning with competing stochastic dynamics in the framework of the modulated phase field method. Analytical results obtained within the mean field theory are compared with computer simulations.     

Structure and dynamics of dislocations in an anisotropic pattern-forming systems

The motion of dislocations in convective roll patterns provides an important wavevector selection mechanism. In this work the structure and velocity of dislocations is calculated near threshold using amplitude equations appropriate for systems with an axial anisotropy. The fact that then the roll pattern has a preferred direction leads to characteristic differences to isotropic systems like Rayleigh-Bénard convection in simple fluids. Furthermore the nucleation process of dislocation pairs is discussed by analyzing the threshold solution that describes the nucleation barrier.     

Generic transient memory formation in disordered systems with noise

Out-of-equilibrium disordered systems may form memories of external driving in a remarkable fashion. The system "remembers" multiple values from a series of training inputs yet "forgets" nearly all of them at long times despite the inputs being continually repeated. Here, learning and forgetting are inseparable aspects of a single process. The memory loss may be prevented by the addition of noise. We identify a class of systems with this behavior, giving as an example a model of non-Brownian suspensions under cyclic shear.     

Method of local statistical equivalence and induced transitions in dynamical systems perturbed by weak noise

We propose a method of local statistical equivalence which in some cases reduces, under condition of small noise intensity, the analysis of the statistical characteristics of noise-induced transitions in dynamical systems to solution of a similar problem in simpler systems. For a system of coupled analytical bistable elements, we found analytical expressions (and compared them with the results of numerical simulation) to estimate the average times of transitions due to changes in the parameter, as counterparts of nonstationary phase transitions of the first and second order. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Lobachevsky State University of Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp 1543–1550, December, 1998.     

The asymptotic dynamics of processes with multiplicative quadratic noise

The mean value of a non-Markovian stochastic process modelled by the rate equation with a multiplicative Ornstein-Uhlenbeck noise is investigated. The stability properties are discussed and the temporal evolution of the mean value and fluctuations of the process are presented. The effective relaxation time of the system is analysed.The research reported in this paper is supported in part by the Polish Academy of Sciences under the contract CPBP 1.1.4. The authors acknowledge the referee for valuable remarks.     

Limiting phase trajectories and dynamic transitions in nonlinear periodic systems

The intensity of energy exchange between parts of periodic nonlinear Frenkel-Kontorova and Klein-Gordon lattices is analyzed based on a concept of limiting phase trajectories introduced earlier. It is demonstrated that, with increasing nonlinearity parameter in these lattices, two dynamic transitions take place successively. The first transition is due to the bifurcation of the lower (with respect to frequency) normal mode because of its instability. It is accompanied by the occurrence of two additional normal modes and the separatrix between them. In this case, after this transition and before it, complete energy exchange between parts of the system is possible. The second transition takes place as a result of merging of the limiting phase trajectory with the separatrix, after which complete energy exchange between parts of the system is impossible. Analytical results are proven by numerical data.     

A comparison between transitions induced by random and periodic fluctuations

It is shown that a certain class of nonlinear systems possesses a unique and stable stationary state when subjected to periodic dichotomous modulations of an external parameter. This result enables us to define a probability density for the system and to characterize its shape and support. We compare this probability density with the one obtained in the case that the external parameter fluctuates randomly like a Markovian dichotomous noise and discuss various fluctuation-induced transition phenomena. The effects of these two types of fluctuations are quite dissimilar: the random fluctuations give rise to a richer behavior. The results are applied to the Freedericksz transition in nematic liquid crystals.Fellow of the University of Texas at Austin.     

Chaotic mixing induced transitions in reaction-diffusion systems

We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection. (c) 2002 American Institute of Physics.     

Diffusive dynamics and periodic orbits of dynamical systems

We show how to organize the set of periodic orbits of dynamical systems exhibiting deterministic diffusion into a cycle expansion giving the diffusion coefficient D. This expansion is used to derive analytically D for a class of such dynamical systems.     

Dissipative quantum dynamics for systems periodic in time

A model of dissipative quantum dynamics (with a nonlinear friction term) is applied to systems periodic in time. The model is compared with the standard approaches based on the Floquet theorem. It is shown that for weak frictions the asymptotic states of the dynamics we propose are the periodic steady states which are usually postulated to be the states relevant for the statistical mechanics of time-periodic systems. A solution to the problem of nonuniqueness of the “quasienergies” is proposed. The implication of a nonlinear evolution for Ludwig's axiomatization is briefly outlined.     

Origin of transient and intermittent dynamics in spatiotemporal chaotic systems

Nonattracting chaotic sets (chaotic saddles) are shown to be responsible for transient and intermittent dynamics in an extended system exemplified by a nonlinear regularized long-wave equation, relevant to plasma and fluid studies. As the driver amplitude is increased, the system undergoes a transition from quasiperiodicity to temporal chaos, then to spatiotemporal chaos. The resulting intermittent time series of spatiotemporal chaos displays random switching between laminar and bursty phases. We identify temporally and spatiotemporally chaotic saddles which are responsible for the laminar and bursty phases, respectively. Prior to the transition to spatiotemporal chaos, a spatiotemporally chaotic saddle is responsible for chaotic transients that mimic the dynamics of the post-transition attractor.     

Pattern dynamics and optimization by reaction diffusion systems

Reaction-diffusion systems show a fast and rather complex response on patterns produced by external space- and/or time-dependent perturbations. For example, one-component autocatalytic reactions rapidly find the loci where the given space-dependent reaction rates have relatively high values by following a kind of Darwinian strategy (combining self-reproduction and diffusion). It is shown that a simulation of this strategy in combination with annealing (decreasing the diffusion rates in time) may be used as an alternative to thermodynamic annealing strategies. Many-component reactions, such as the light-sensitive Belousov-Zhabotinsky reaction, show a more complex response to patterns impressed by illumination, for example. The response behavior and possible applications to dynamic information processing are discussed.     

Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing

It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. 88, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. We explain this as an effect of periodic forces. Thus, we give qualitative explanations for how stable reversal phenomena appear, which seem to arise from the mixing function of the periodic force and the noise. The output period and three velocities (normal, positive and negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity. Electronic supplementary material Supplementary Online Material     

Critical behavior and 1/f noise in lumped systems undergoing two phase transitions

It is shown that the interaction of order parameters when subcritical and supercritical phase transitions take place simultaneously may result in a self-organized critical state and cause a 1/f ^α fluctuation spectrum, where 1≤α≤2. Such behavior is inherent in potential and nonpotential systems of nonlinear Langevin equations. A numerical analysis of the solutions to the proposed systems of stochastic differential equations showed that the solutions correlate with fractional integration and differentiation of white noise. The general behavior of such a system has features in common with self-organized criticality.     

Stochastic resonance induced by bounded noise and periodic signal in an asymmetric bistable system

Stochastic resonance (SR) induced by the bounded noise in an asymmetric bistable system is investigated. Based on stochastic simulation, the signal power amplification SPA$SPA$ is derived for the case of the additive modulated signal. The simulation results indicate that: (1) the SR phenomenon is weakened by the asymmetry of the system; (2) increasing the frequency ω$\omega$ restricts the SR of the system; (3) there is an optimum value of amplitude A$A$ matching the strongest SR phenomenon.     

Dynamical suppression of 1/f noise processes in qubit systems

We investigate the capability of dynamical decoupling techniques to reduce decoherence from a realistic environment generating 1/f noise. The predominance of low frequency modes in the noise profile allows for decoherence scenarios where relatively slow control rates suffice for a drastic improvement. However, the actual figure of merit is very sensitive to the details of the dynamics, with decoupling performance which may deteriorate for non-Gaussian noise and/or high frequency working points. Our results are promising for robust solid-state qubits and beyond.     

External noise and the origin and dynamics of structure in convectively unstable systems

Some basic concepts and earlier work on external noise and the convectively unstable Ginzburg-Landau equation are reviewed, and some of the ideas presented in the earlier work are investigated further and extended. In particular, further consideration is given to convective chaos-chaos which only occurs in a moving frame of reference; and slugs-localized structures which are surrounded by a stable stationary state. Some new results on secondary convective instabilities and on periodic systems with a spatially varying instability are discussed. Work on the coupled Ginzburg-Landau equation is reviewed. Actual physical systems are discussed.     

分子尺度不对称体系中热噪声引发的单向输运

不对称体系中的单向输运现象广泛存在于自然界中,宏观理论认为,体系的空间反演不对称性和有一定自相关时间的外加振动是引发单向运动的两个必要条件,因而一般认为,在分子尺度的不对称体系中,仅有热噪声也还是不会引发单向运动的。随着实验技术和计算手段的发展,有迹象表明,在分子尺度的不对称体系中,即使没有外加振动也会有单向运动产生。文章介绍在分子尺度由体系热噪声引发的不对称体系中,单向输运的理论进展,以及产生这种单向输运的条件。