Fluctuation and Noise Letters

We examine the kinetics of relaxation processes in dynamical models with the fixed number of Brownian particles and with the constant source of the particles. The former evolves to the zero equilibrium distribution, while the latter develops to the stationary state with non-equilibrium distribution and non-zero stationary flow. We show that the relaxation times in the both systems coincide. This is valid for an arbitrary noise intensity and shape of potential profile. In addition we propose to modify the well-known Kramers' approach for the determination of relaxation time. The modified approach is valid for an arbitrary height of potential barriers.

We investigate the way in which noise destroys phase synchronization in chaotic systems. Two cases are considered: where the route to synchronization involves frequency locking; and where it occurs via a suppression of the natural dynamics. We show that, just as in the case of synchronization of periodic oscillations, noise induces a new motion whose coherence depends non-monotonically (resonantly) on the noise intensity.

The response of a bistable vertical cavity surface emitting laser to a periodic input signal display all the hallmarks of stochastic resonance. Accurate experimental measurements show the occurrence of noise-induced frequency and effective phase locking characterized by a plateau of the mean output switching rate and a minimum of the relative–phase diffusion constant, respectively.

The control of the low-frequency fluctuations exhibited by two mutually coupled semiconductor lasers is studied experimentally and numerically. We observe that coupling enhances the response of the system to a weak periodic modulation of the injection current of one of the lasers, leading to a highly efficient entrainment of the synchronized low-frequency power dropouts to the external periodic driving. We compare the quality of the entrainment with the one obtained in a single semiconductor laser with optical feedback, showing the beneficial role of coupling in this pursuit. The experimental observations are satisfactorily reproduced by numerical simulations of a set of coupled delay-differential rate equations.

We consider here the dynamics and stochastic theory of swarms of self-propelled particles. Driving is modelled by the Schienbein-Gruler expression for negative friction. Global coupling is introduced by linear forces attracting to the center of mass and to the mean velocity of the swarm. Solutions for the stationary distribution of swarms are given which represent: (i) synchronized translation of the swarm with small fluctuations around its center of mass and, (ii) synchronized rotations around the center of mass which is at rest or slowly moving.

Biological motion and human traffic require energy supply from external sources. We develop here a model for the dynamics of driven entities which includes hydrodynamic interactions in order to adapt the model to the dynamics of swarms moving in dense fluids. Our entities have the ability to use the energy contained in an internal energy depot or an external energy inflow for the acceleration of motion. As a prototype of such entities we study Brownian particles having the ability to take up energy from their environment, to store it in an internal energy depot and to convert internal energy into kinetic energy. The motion of the particles is described by Langevin equations which include a dissipative force term resulting from the driving and equations for the dynamics of the depot. The hydrodynamic interactions are modeled by an Oseen-type tensorial force. It is shown that hydrodynamic interactions lead to the synchronization of the directions of motion leading to several new collective modes of the dynamics, including spontaneous rotations of the swarm.

We study numerically the stationary solutions of the Fokker-Planck equation for the FitzHugh-Nagumo model with additive noise. In the parameter regimes where the deterministic model is excitable we find various sets of maxima, minima, and saddle points of the stationary probability distribution depending on the noise intensity and separation of the time scales between activator and inhibitor.

There is strong evidence for couplings between the five oscillatory processes that characterize the human blood distribution system. In particular, mutual modulation and/or episodes of synchronization between the oscillations may be observed for a variety of different cardiovascular signals. Such phenomena can reveal information about the nature and strength of the couplings, which in turn reflect the state of the organism in sickness or in health. Earlier work on the inter-oscillator interactions is reviewed briefly, and some recent research on cardio-respiratory synchronization is considered in more detail. It is reported that, for rats, the directionality of the cardiorespiratory coupling reverses under anæsthesia. The interplay between synchronization and modulation is discussed and their coexistence in the cardiorespiratory interaction is illustrated.

We study the time evolution of two ecosystems in the presence of external noise and climatic periodical forcing by a generalized Lotka-Volterra (LV) model. In the first ecosystem, composed by two competing species, we find noise induced phenomena such as: (i) quasi deterministic oscillations, (ii) stochastic resonance, (iii) noise delayed extinction and (iv) spatial patterns. In the second ecosystem, composed by three interacting species (one predator and two preys), using a discrete model of the LV equations we find that the time evolution of the spatial patterns is strongly dependent on the initial conditions of the three species.

In this paper we present a simple way of chaos detection in forced dynamical systems. The method exploits a synchronization phenomenon of a set of identical mechanical systems driven by the same external excitation. The properties of this synchronization allow for easy and fast detection of chaos in dynamical systems in a wide range of system parameters. The presented approach can be successfully applied both for discrete-time and continuous-time systems. Some applications of the method are presented.

The beneficial role fluctuations can play in the process of phase synchronization is analyzed in terms of a model with binary input and output signals. Special attention is paid to the relation between noise-induced phase synchronization and the well-known phenomenon of stochastic resonance. Analytic predictions are compared with experimental data from a vertical cavity surface emitting laser. Various measures for aperiodic stochastic resonance, frequency entrainment and stochastic phase synchronization reveal a satisfactory agreement between theory and experiment.

The effect of anomalously enhanced quality factor (small damping decrement) is demonstrated and explained by phase dependence of interaction with particles of medium. The dependence of quality factor on the temperature of the medium is described. The most suitable noise intensity is determined. Molecular dynamics simulations qualitatively confirm the results. The size and mass of real bio-molecules, the frequency, amplitude of their oscillations and the temperature of living body are suitable conditions for this effect.

We study numerically the behavior of the autocorrelation function (ACF) and the power spectrum of spiral attractors without and in the presence of noise. It is shown that the ACF decays exponentially and has two different time scales. The rate of the ACF decrease is defined by the amplitude fluctuations on small time intervals, i.e., when τ < τ_cor, and by the effective diffusion coefficient of the instantantaneous phase on large time intervals. It is also demonstrated that the ACF in the Poincare map also decreases according to the exponential law exp(- λ⁺ k), where λ⁺ is the positive Lyapunov exponent. The obtained results are compared with the theory of fluctuations for the Van der Pol oscillator.

A general concept is presented which allows of setting up mathematical models for stochastic and quasi deterministic dynamic processes in social systems.

The basis of this concept is the master equation for the probability distribution over appropriately chosen personal and material macrovariables of the society. The probabilistic transition rates depend on motivation potentials governing the decisions and actions of the social agents. The transition from the probability distribution to quasi-meanvalues leads to in general nonlinear coupled differential equations for the macrovariables of the chosen social sector. Up to now several models about population dynamics, collective political opinion formation, dynamics of economic processes and the formation of settlements have been published.

We address the problem of the chaotic transport of particles in an asymmetric periodic ratchet potential. We consider the case of deterministic rocking ratchets and analyze different regimes for the amplitude of rocking. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. We compare the dynamics of point particles and extended objects under the action of the ratchet potential and show that we can segregate particles according to its size.

The fractal properties of the Lattice Limit Cycle model are explored when the process is realized on a 2-dimensional square lattice support via Monte Carlo Simulations. It is shown that the structure of the steady state presents inhomogeneous fluctuations in the form of domains of identical particles. The various domains compete with one another via their borders which have self-similar, fractal structure. The fractality is more prominent, (fractal dimensions d_f < 2), when the parameter values are near the critical point where the Hopf bifurcation occurs. As the distance from the Hopf bifurcation increases in the parameter space the system becomes more homogeneous and the fractal dimension tends to the value d_f = 2.

We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in the Harper equation, including the renormalization strange set (known as the orchid) in the generalized Harper equation; and (ii) self-similarities in the correlations of strange nonchaotic attractors, barrier billiards, and quantum two-level systems.