Linear stability analysis for bifurcations in spatially extended systems with fluctuating control parameter

Tuning limit cycles with a noise: Survival and collapse

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equation of motion for slow dynamics after appropriate averaging over the fast motion. The equation for slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

Heterogeneity-stabilized homogeneous states in driven media

Understanding the relationship between symmetry breaking, system properties, and instabilities has been a problem of longstanding scientific interest. Symmetry-breaking instabilities underlie the formation of important patterns in driven systems, but there are many instances in which such instabilities are undesirable. Using parametric resonance as a model process, here we show that a range of states that would be destabilized by symmetry-breaking instabilities can be preserved and stabilized by the introduction of suitable system asymmetry. Because symmetric states are spatially homogeneous and asymmetric systems are spatially heterogeneous, we refer to this effect as heterogeneity-stabilized homogeneity. We illustrate this effect theoretically using driven pendulum array models and demonstrate it experimentally using Faraday wave instabilities. Our results have potential implications for the mitigation of instabilities in engineered systems and the emergence of homogeneous states in natural systems with inherent heterogeneities.

Impact of correlated noise in an energy depot model

Based on the depot model of the motion of active Brownian particles (ABPs), the impact of cross-correlated multiplicative and additive noises has been investigated. Using a nonlinear Langevin approach, we discuss a new mechanism for the transport of ABPs in which the energy originates from correlated noise. It is shown that the correlation between two types of noise breaks the symmetry of the potential to generate motion of the ABPs with a net velocity. The absolute maximum value of the mean velocity depends on correlated noise or multiplicative noise, whereas a monotonic decrease in the mean velocity occurs with additive noise. In the case of no correlation, the ABPs undergo pure diffusion with zero mean velocity, whereas in the case of perfect correlation, the ABPs undergo pure drift with zero diffusion. This shows that the energy stemming from correlated noise is primarily converted to kinetic energy of the intrawell motion and is eventually dissipated in drift motion. A physical explanation of the mechanisms for noise-driven transport of ABPs is derived from the effective potential of the Fokker-Planck equation.

Landscape-inversion phase transition in dipolar colloids: tuning the structure and dynamics of 2D crystals

We study the 2D crystalline phases of paramagnetic colloidal particles with dipolar interactions and constrained on a periodic substrate. Combining theory, simulation, and experiments, we demonstrate a new scenario of first-order phase transitions that occurs via a complete inversion of the energy landscape, featuring nonconventional properties that allow for (i) tuning of crystal symmetry, (ii) control of dynamical properties of different crystalline orders via tuning of their relative stability with an external magnetic field, (iii) an equivalent but independent control of the same dynamic properties via temporal modulations of that field, and (iv) nonstandard phase-ordering kinetics involving spontaneous formation of transient metastable domains.

Noise-induced threshold shift and pattern formation in electroconvection by controlling characteristic time scales

We report external noise-induced threshold shift and pattern formation in ac-driven electroconvection (EC) in nematic liquid crystals. We investigate the noise intensity dependence of the threshold and determine the relationship f(c)(*)=hf(cd)(a) (α ∼ 1.4, h ∼ 0.1) between the characteristic cutoff frequency (f(c)(*)) of the noise and the characteristic ac frequency (f(cd)) of EC. For f(c)>f(c)(*), the noise contributes to stabilizing EC (i.e., upward threshold shift), whereas for f(c)<f(c)(*), it contributes to destabilizing EC (i.e., downward threshold shift). For f(c)=f(c)(*), the noise makes no contribution to the threshold shift. However, the characteristic wavelength of EC strongly depends on the noise intensity but not on f(c).

Control of the Hopf-Turing transition by time-delayed global feedback in a reaction-diffusion system

Application of time-delayed feedback is a practical method of controlling bifurcations in reaction-diffusion systems. It has been shown that for a suitable feedback strength, time delay beyond a threshold may induce spatiotemporal instabilities. For an appropriate parameter space with differential diffusivities of the activator-inhibitor species, delayed feedback may generate Turing instability via a Hopf-Turing transition, resulting in stationary patterns. This is explored by a theoretical scheme in a photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system where the delayed feedback is externally tuned by photoillumination intensity. Our analytical results corroborate with direct numerical simulations.

Galerkin analysis of light-induced patterns in the chlorine dioxide-iodine-malonic acid reaction-diffusion system

The photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system has been an experimental paradigm for the study of Turing pattern over the last several years. When subjected to illumination of varied intensity by visible light the patterns undergo changes from spots to stripes, vice versa, and their mixture. We carry out a nonlinear analysis of the underlying model in terms of a Galerkin scheme with finite number of modes to explore the nature of the stability and existence of various modes responsible for the type and crossover of the light-induced patterns.

Nonlinear dynamics of finite perturbation: collapse and revival of spatial patterns

A full-scale nonlinear stability analysis is performed on a reaction-diffusion system that includes a cubic polynomial source term and Cattaneo's modification of Fick's law of diffusion. This modification incorporates the effect of a small, finite relaxation time of flux at the macroscopic level of the description of the process. While linear stability analysis predicts the decay of small wavelength perturbations on a homogeneous steady state for large reaction time, consideration of finite perturbations leads to a spatiotemporal instability, resulting in an interesting phenomenon of periodic collapse and revival of spatial patterns. This instability is relaxation (time) driven, and the time period is determined by self-sustaining oscillations due to the limit cycle of the underlying dynamics. The nonlinear dynamics of finite perturbations may thus be generically different from what is expected from a linear stability analysis.

Instability and pattern formation in reaction-diffusion systems: A higher order analysis

We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.

Resonant-pattern formation induced by additive noise in periodically forced reaction-diffusion systems

We report frequency-locked resonant patterns induced by additive noise in periodically forced reaction-diffusion Brusselator model. In the regime of 2:1 frequency-locking and homogeneous oscillation, the introduction of additive noise, which is colored in time and white in space, generates and sustains resonant patterns of hexagons, stripes, and labyrinths which oscillate at half of the forcing frequency. Both the noise strength and the correlation time control the pattern formation. The system transits from homogeneous to hexagons, stripes, and to labyrinths successively as the noise strength is adjusted. Good frequency-locked patterns are only sustained by the colored noise and a finite time correlation is necessary. At the limit of white noise with zero temporal correlation, irregular patterns which are only nearly resonant come out as the noise strength is adjusted. The phenomenon induced by colored noise in the forced reaction-diffusion system is demonstrated to correspond to noise-induced Turing instability in the corresponding forced complex Ginzburg-Landau equation.

Front propagation sustained by additive noise

The effect of noise in a motionless front between a periodic spatial state and an homogeneous one is studied. Numerical simulations show that noise induces front propagation. From the subcritical Swift-Hohenberg equation with noise, we deduce an adequate equation for the envelope and the core of the front. The equation of the core of the front is characterized by an asymmetrical periodic potential plus additive noise. The conversion of random fluctuations into direct motion of the core of the front is responsible of the propagation. We obtain an analytical expression for the velocity of the front, which is in good agreement with numerical simulations.

Regular patterns in dichotomically driven activator-inhibitor dynamics

We investigate Turing pattern formation in the presence of additive dichotomous fluctuations in the context of an extended system with diffusive coupling and FitzHugh-Nagumo kinetics. The fluctuations vary in space and/or time. Depending on the realization of the dichotomous switching the system is, at a given time (for spatial disorder at a given position) in one of two possible excitable dynamical regimes. Each of the two excitable dynamics for itself does not support pattern formation. With proper dichotomous fluctuations, however, the homogeneous steady state is destabilized via a Turing instability. We investigate the influence of different switching rates (different correlation length of the spatial disorder) on pattern formation. We find three distinct mechanisms: For slow switching existing boundaries become unstable, for high rates the system exhibits "effective bistability" which allows for a Turing instability. For medium rates the fluctuations create spatial structures via a new mechanism where the influence of the fluctuations is twofold. First they produce local inhomogeneities, which then grow (again caused by fluctuations) until the whole space is covered. Utilizing a nonlinear map approach we show bistability of a period-one and a period-two orbit being associated with the steady homogeneous and the Turing pattern state, respectively. Finally, for purely static dichotomous disorder we find destabilization of homogeneous steady states for finite nonzero correlation length of the disorder resulting again in Turing patterns.

Quantum escape kinetics over a fluctuating barrier

The escape rate of a particle over a fluctuating barrier in a double-well potential exhibits resonance at an optimum value of correlation time of fluctuation. This has been shown to be important in several variants of kinetic model of chemical reactions. We extend the analysis of this phenomenon of resonant activation to quantum domain to show how quantization significantly enhances resonant activation at low temperature due to tunneling.

Effect of noise on defect chaos in a reaction-diffusion model

The influence of noise on defect chaos due to breakup of spiral waves through Doppler and Eckhaus instabilities is investigated numerically with a modified Fitzhugh-Nagumo model. By numerical simulations we show that the noise can drastically enhance the creation and annihilation rates of topological defects. The noise-free probability distribution function for defects in this model is found not to fit with the previously reported squared-Poisson distribution. Under the influence of noise, the distributions are flattened, and can fit with the squared-Poisson or the modified-Poisson distribution. The defect lifetime and diffusive property of defects under the influence of noise are also checked in this model.

Ising-Bloch transition for the parametric Ginzburg-Landau equation with rapidly varying perturbations

We study the effects of rapid periodic and stochastic modulations of parameters in systems described by the complex parametric Ginzburg-Landau equation. Amplitude equations, which govern the dynamics of the field averaged over the rapid modulations, are derived. For temporal modulations of the linear detuning the threshold for the transition from Ising to Bloch walls is shifted depending on the strength of the perturbation. In contrast to this, rapid perturbations of the linear gain lead only to a decrease of the amplitude of both wall types leaving the bifurcation point of the Ising-Bloch transition unchanged. Stochastic perturbations of the detuning lead to a Brownian motion of the Bloch wall beyond bifurcation where the velocity is given analytically. All theoretical predictions are confirmed by numerical simulations of the full stochastic Ginzburg-Landau equation.

Additive noise induces front propagation

The effect of additive noise on a static front that connects a stable homogeneous state with an also stable but spatially periodic state is studied. Numerical simulations show that noise induces front propagation. The conversion of random fluctuations into direct motion of the front's core is responsible of the propagation; noise prefers to create or remove a bump, because the necessary perturbations to nucleate or destroy a bump are different. From a prototype model with noise, we deduce an adequate equation for the front's core. An analytical expression for the front velocity is deduced, which is in good agreement with numerical simulations.

Noise-induced instability: an approach based on higher-order moments

Noise-induced transitions in the organization of systems far from equilibrium have been of vital interest. Although the effects of additive and multiplicative noise have been widely studied, it is only the multiplicative noise that can be dealt with within the scope of a linear analysis of first moments of the spatiotemporal perturbations, by the application of Novikov's theorem. For the case of additive noise, the corresponding straightforward linear analysis of the first moment throws no light on the effect of the noise on stability conditions. We propose here a simple approach based on higher-order moments to show how additive noise can give rise to noise-induced instability in spatially extended systems, at times leading to pattern formation. Our theoretical analysis is corroborated by numerical simulations on two simple one-component reaction-diffusion systems in two dimensions.

Statistics of defect-mediated turbulence influenced by noise

The influence of white noise on defect-mediated turbulence which is modeled by the complex Ginzburg-Landau equation is investigated. We show that the dynamics of defects in the noise-driven spatiotemporal chaos can be described by a simple statistical model. The noise enhances significantly the ability of the turbulent background to advocate new defects with a constant rate, and at the same time it increases the vanishing of defects in the system by introducing an additional annihilation rate that is proportional to the number of defects. A universal probability distribution function is derived for the number of defect pairs.

Intrinsic noise-induced phase transitions: beyond the noise interpretation

We discuss intrinsic noise effects in stochastic multiplicative-noise partial differential equations, which are qualitatively independent of the noise interpretation (Itô vs Stratonovich), in particular in the context of noise-induced ordering phase transitions. We study a model which, contrary to all cases known so far, exhibits such ordering transitions when the noise is interpreted not only according to Stratonovich, but also to Itô. The main feature of this model is the absence of a linear instability at the transition point. The dynamical properties of the resulting noise-induced growth processes are studied and compared in the two interpretations and with a reference Ginzburg-Landau-type model. A detailed discussion of a different numerical algorithm valid for both interpretations is also presented.

Noise-driven mechanism for pattern formation

We extend the mechanism for noise-induced phase transitions proposed by Ibañes et al. [Phys. Rev. Lett. 87, 020601 (2001)] to pattern formation phenomena. In contrast with known mechanisms for pure noise-induced pattern formation, this mechanism is not driven by a short-time instability amplified by collective effects. The phenomenon is analyzed by means of a modulated mean field approximation and numerical simulations.

Effect of colored noises on spatiotemporal chaos in the complex Ginzburg-Landau equation

The effect of colored noises, which are correlated both in space and time, on spatiotemporal chaos in the complex Ginzburg-Landau equation has been studied numerically. The correlations of spatiotemporal patterns as a function of characteristics of noise were calculated. We found that there exists an optimal correlation length of noise where the system establishes its maximal spatial correlation; a small temporal correlation of noise corresponds to a larger correlation length in the system; and that an increase of noise intensity enhances the spatial correlation of the spatiotemporal patterns. Besides, the frequency of temporal correlation function, which is a complicated oscillation, also depends on properties of the noise.

Adiabatic elimination and reduced probability distribution functions in spatially extended systems with a fluctuating control parameter

We obtain the stationary probability distribution functions of the order parameter near onset for the one-dimensional real Ginzburg-Landau and Swift-Hohenberg equations with a fluctuating control parameter. A perturbative expansion in the intensity of the fluctuations leads to a hierarchy of Fokker-Planck equations for conditional probability distribution functions that relate components of the order parameter that evolve in different time scales. Successive integration leads to a Fokker-Planck equation for the slowest mode, which we solve analytically for the models studied. In all cases, the probability distribution function above onset is of the form P(A0) approximately A(delta)(0)e(-gammaA20), where A0 is the slow component of the order parameter and the values of delta and gamma depend explicitly on the intensity of the fluctuations. Knowledge of P(A0) allows the calculation of an effective bifurcation threshold and of the moments of A0 above threshold.

Second-order stochastic leapfrog algorithm for multiplicative noise brownian motion

A stochastic leapfrog algorithm for the numerical integration of Brownian motion stochastic differential equations with multiplicative noise is proposed and tested. The algorithm has a second-order convergence of moments in a finite time interval and requires the sampling of only one uniformly distributed random variable per time step. The noise may be white or colored. We apply the algorithm to a study of the approach towards equilibrium of an oscillator coupled nonlinearly to a heat bath and investigate the effect of the multiplicative noise (arising from the nonlinear coupling) on the relaxation time. This allows us to test the regime of validity of the energy-envelope approximation method.

Lattice-independent approach to thermal phase mixing

We show how to achieve lattice-spacing-independent results in numerical simulations of finite-temperature stochastic scalar field theories. We generalize a previous approach by obtaining results which are independent of the renormalization scale. As an application of our method, we examine thermal phase mixing in the context of Ginzburg-Landau models with short-range interactions. In particular, we obtain the lattice-spacing and renormalization-scale-independent critical value of the control parameter which determines the free-energy barrier between the two low-temperature phases. We also propose a simple procedure to extract the critical value of control parameters for different choices of lattice spacing.

Scaling of lyapunov exponents of coupled chaotic systems

We develop a statistical theory of the coupling sensitivity of chaos. The effect was first described by Daido [Prog. Theor. Phys. 72, 853 (1984)]; it appears as a logarithmic singularity in the Lyapunov exponent in coupled chaotic systems at very small couplings. Using a continuous-time stochastic model for the coupled systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered, too. The analytical results are confirmed by numerical simulations.

Enhanced pulse propagation in nonlinear arrays of oscillators

The propagation of a pulse in a nonlinear array of oscillators is influenced by the nature of the array and by its coupling to a thermal environment. For example, in some arrays a pulse can be speeded up while in others a pulse can be slowed down by raising the temperature. We begin by showing that an energy pulse (one dimension) or energy front (two dimensions) travels more rapidly and remains more localized over greater distances in an isolated array (microcanonical) of hard springs than in a harmonic array or in a soft-springed array. Increasing the pulse amplitude causes it to speed up in a hard chain, leaves the pulse speed unchanged in a harmonic system, and slows down the pulse in a soft chain. Connection of each site to a thermal environment (canonical) affects these results very differently in each type of array. In a hard chain the dissipative forces slow down the pulse while raising the temperature speeds it up. In a soft chain the opposite occurs: the dissipative forces actually speed up the pulse, while raising the temperature slows it down. In a harmonic chain neither dissipation nor temperature changes affect the pulse speed. These and other results are explained on the basis of the frequency vs energy relations in the various arrays.

Noise-induced phase separation: mean-field results

We present a study of a phase-separation process induced by the presence of spatially correlated multiplicative noise. We develop a mean-field approach suitable for conserved-order-parameter systems and use it to obtain the phase diagram of the model. Mean-field results are compared with numerical simulations of the complete model in two dimensions. Additionally, a comparison between the noise-driven dynamics of conserved and nonconserved systems is made at the level of the mean-field approximation.

Recent results on multiplicative noise

Recent developments in the analysis of Langevin equations with multiplicative noise (MN) are reported. In particular, we (i) present numerical simulations in three dimensions showing that the MN equation exhibits, like the Kardar-Parisi-Zhang (KPZ) equation, both a weak coupling fixed point and a strong coupling phase, supporting the proposed relation between MN and KPZ; (ii) present a dimensional and mean-field analysis of the MN equation to compute critical exponents; (iii) show that the phenomenon of the noise-induced ordering transition associated with the MN equation appears only in the Stratonovich representation and not in the Ito one; and (iv) report the presence of a first-order-like phase transition at zero spatial coupling, supporting the fact that this is the minimum model for noise-induced ordering transitions.