Stochastic spatiotemporal intermittency and noise-induced transition to an absorbing phase

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.

Front depinning by deterministic and stochastic fluctuations: A comparison

Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.

Internal noise and system size effects induce nondiffusive kink dynamics

We investigate the effects of inherent fluctuations and system size in the dynamics of domain between uniform symmetric states. In the case of monotonous kinks, this dynamics is characterized by exhibiting nonsymmetric random walks, being attracted to the system borders. For nonmonotonous interface, the dynamics is replaced by a hopping dynamic. Based on bistable universal models, we characterize the origin of these unexpected dynamics through use of the stochastic kinematic laws for the interface position and the survival probability. Numerical simulations show a quite good agreement with the theoretical predictions.

Crisis, unstable dimension variability, and bifurcations in a system with high-dimensional phase space: coupled sine circle maps

The phenomenon of crisis in systems evolving in high-dimensional phase space can show unexpected and interesting features. We study this phenomenon in the context of a system of coupled sine circle maps. We establish that the origins of this crisis lie in a tangent bifurcation in high dimensions, and identify the routes that lead to the crisis. Interestingly, multiple routes to crisis are seen depending on the initial conditions of the system, due to the high dimensionality of the space in which the system evolves. The statistical behavior seen in the phase diagram of the system is also seen to change due to the dynamical phenomenon of crisis, which leads to transitions from nonspreading to spreading behavior across an infection line in the phase diagram. Unstable dimension variability is seen in the neighborhood of the infection line. We characterize this crisis and unstable dimension variability using dynamical characterizers, such as finite-time Lyapunov exponents and their distributions. The phase diagram also contains regimes of spatiotemporal intermittency and spatial intermittency, where the statistical quantities scale as power laws. We discuss the signatures of these regimes in the dynamic characterizers, and correlate them with the statistical characterizers and bifurcation behavior. We find that it is necessary to look at both types of correlators together to build up an accurate picture of the behavior of the system.

Spatial pattern formation in external noise: theory and simulation

Spatial pattern formation in fluctuating media is researched analytically from the point of view of the order parameters concept. A reaction-diffusion system with external noise is considered as a model of such media. Stochastic equations for unstable mode amplitudes (order parameters), the dispersion equation for averaged amplitudes of unstable modes, and the Fokker-Planck equation for the order parameters are obtained. The theory developed makes it possible to analyze different noise-induced effects including the variation of boundaries of ordering and disordering phase transitions depending on the parameters of external noise.

Universal shape law of stochastic supercritical bifurcations: theory and experiments

A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional (1D) systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement.

Critical wetting of a class of nonequilibrium interfaces: a mean-field picture

A self-consistent mean-field method is used to study critical wetting transitions under nonequilibrium conditions by analyzing Kardar-Parisi-Zhang (KPZ) interfaces in the presence of a bounding substrate. In the case of positive KPZ nonlinearity a single (Gaussian) regime is found. On the contrary, interfaces corresponding to negative nonlinearities lead to three different regimes of critical behavior for the surface order parameter: (i) a trivial Gaussian regime, (ii) a weak-fluctuation regime with a trivially located critical point and nontrivial exponents, and (iii) a highly nontrivial strong-fluctuation regime, for which we provide a full solution by finding the zeros of parabolic-cylinder functions. These analytical results are also verified by solving numerically the self-consistent equation in each case. Analogies with and differences from equilibrium critical wetting as well as nonequilibrium complete wetting are also discussed.

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.

Synchronized chaotic intermittent and spiking behavior in coupled map chains

We study phase synchronization effects in a chain of nonidentical chaotic oscillators with a type-I intermittent behavior. Two types of parameter distribution, linear and random, are considered. The typical phenomena are the onset and existence of global (all-to-all) and cluster (partial) synchronization with increase of coupling. Increase of coupling strength can also lead to desynchronization phenomena, i.e., global or cluster synchronization is changed into a regime where synchronization is intermittent with incoherent states. Then a regime of a fully incoherent nonsynchronous state (spatiotemporal intermittency) appears. Synchronization-desynchronization transitions with increase of coupling are also demonstrated for a system resembling an intermittent one: a chain of coupled maps replicating the spiking behavior of neurobiological networks.

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.

Stochastic theory of synchronization transitions in extended systems

We propose a general Langevin equation describing the universal properties of synchronization transitions in extended systems. By means of theoretical arguments and numerical simulations we show that the proposed equation exhibits, depending on parameter values: (i) a continuous transition in the bounded Kardar-Parisi-Zhang universality class, with a zero largest Lyapunov exponent at the critical point; (ii) a continuous transition in the directed percolation class, with a negative Lyapunov exponent, or (iii) a discontinuous transition (that is argued to be possibly just a transient effect). Cases (ii) and (iii) exhibit coexistence of synchronized and unsynchronized phases in a broad (fuzzy) region. This reproduces almost all of the reported features of synchronization transitions, providing a unified theoretical framework for the analysis of synchronization transitions in extended systems.

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.

Critical exponents of directed percolation measured in spatiotemporal intermittency

An experimental system showing a transition to spatiotemporal intermittency is presented. It consists of a ring of hundred oscillating ferrofluidic spikes. Four of five of the measured critical exponents of the system agree with those obtained from a theoretical model of directed percolation.

Nonequilibrium wetting transitions with short range forces

We analyze within mean-field theory as well as numerically a Kardar-Parisi-Zhang equation that describes nonequilibrium wetting. Both complete and critical wettitng transitions were found and characterized in detail. For one-dimensional substrates the critical weting temperature is depressed by fluctuations. In addition, we have investigated a region in the space of parameters (temperature and chemical potential) where the wet and nonwet phases coexist. Finite-size scaling analysis of the interfacial detaching times indicates that the finite coexistence region survives in the thermodynamic limit. Within this region we have observed (stable or very long lived) structures related to spatiotemporal intermittency in other systems. In the interfacial representation these structures exhibit perfect triangular (pyramidal) patterns in one dimension (two dimensions), which are characterized by their slope and size distribution.

Spatiotemporal intermittency and scaling laws in inhomogeneous coupled map lattices

We study the phenomenon of intermittency in an inhomogeneous lattice of coupled maps where the inhomogeneity appears in the form of different values of the map parameter at adjacent sites. This system exhibits spatiotemporal intermittency as well as purely spatial intermittency accompanied by temporal periodicity in different regions of the parameter space. Both types of intermittency appear as a result of bifurcations of codimension two in such systems. We identify the types of bifurcations that are seen. The intermittency near the bifurcation points and lines is associated with power-law distributions for the laminar lengths. The scaling laws for the laminar length distributions are obtained. Two distinct types of scaling behavior characterized by power laws with exponents that fall in two distinct ranges can be seen in the neighborhood of codimension-two bifurcation points. Additionally we find two crossover exponents.