Noise-induced switching in dynamics of oscillating populations coupled by migration.

The problem of identifying the sources of switching in the dynamics of nonlinear coupled systems and their mathematical prediction is considered. We study a metapopulation system formed by two oscillating subpopulations coupled by mutual migration. For this model, parametric zones of mono-, bi-, and tri-rhythmicity with the coexistence of regular and chaotic attractors are revealed. The effects of random perturbations in the migration intensity parameter are studied both by methods of statistical analysis of the results of direct numerical simulation and by using the analytical technique of stochastic sensitivity. Noise-induced transitions between anti- and in-phase synchronization modes, as well as between order and chaos, are being studied. Here, the role of transient chaotic attractors and their fractal basins is discussed.

Noise-driven bursting birhythmicity in the Hindmarsh-Rose neuron model.

The stochastic Hindmarsh-Rose model is studied in the parameter region where two bursting limit cycles of different types coexist. We show that under the influence of noise, transitions between basins of attractions appear, which generates stochastic bursting oscillations of mixed modes. The formation of this new regime is accompanied by anti-coherence and coherence resonances as well as by the transition to chaos. We investigate the probabilistic mechanism of the noise-driven bursting birhythmicity using the stochastic sensitivity functions and confidence domains method.

Noise-induced formation of heterogeneous patterns in the Turing stability zones of diffusion systems.

We study a phenomenon of stochastic generation of waveform patterns for reaction-diffusion systems in the Turing stability zone where the homogeneous equilibrium is a single attractor. In this analysis, we use a distributed variant of the Selkov glycolytic model with diffusion and random forcing. It is shown that in the Turing stability zone, random disturbances can induce a diversity of metastable spatial patterns with different waveforms. We carry out the parametric analysis of statistical characteristics of evolution of these patterns, and reveal the dominant patterns in the stochastic flow of mixed spatial structures.

How noise can generate calcium spike-type oscillations in deterministic equilibrium modes.

Stochastic excitability of spiking oscillatory regimes in the calcium kinetics is studied on the basis of the Li-Rinzel conceptual model. The probabilistic mechanisms of the noise-induced generation of large-amplitude oscillations in parametric zones, where the original deterministic model has only stable equilibria, are investigated numerically and analytically. A parametric statistical description of interspike intervals is curried out and the phenomenon of coherence resonance is discussed. For the analytical study of the stochastic excitement, the confidence domain method using a stochastic sensitivity technique is applied. In this analysis, a key role of mutual arrangement of the confidence ellipses and separatrices detaching the sub- and supercritical regions is demonstrated. It is shown that in the Li-Rinzel model such separatrices are the stable manifolds of the saddle equilibria and the transient semiattractors.

Analysis of stochastic dynamics in a multistable logistic-type epidemiological model.

Motivated by the important problem of analyzing and predicting the spread of epidemics, we propose and study a discrete susceptible-infected model. This logistic-type model accounts such significant parameters as the rate of infection spread due to contacts, mortality caused by disease, and the rate of recovery. We present results of the bifurcation analysis of regular and chaotic survival regimes for interacting susceptible and infected subpopulations. Parametric zones of multistability are found and basins of coexisting attractors are determined. We also discuss the particular role of specific transients. In-phase and anti-phase synchronization in the oscillations of the susceptible and infected parts of the population is studied. An impact of inevitably present random disturbances is studied numerically and by the analytical method of confidence domains. Various mechanisms of noise-induced extinction in this epidemiological model are discussed.

Slow-fast oscillatory dynamics and phantom attractors in stochastic modeling of biochemical reactions.

A problem of the probabilistic analysis of stochastic phenomena in slow-fast dynamical systems modeling biochemical reactions is considered. We study how multiplicative noise induces systematic shifts of probabilistic distributions and forms "phantom" attractors in nonlinear enzymatic models. The mathematical analysis of the underlying probabilistic mechanism of such stochastic transformations is performed by the "freeze-and-average" method. Our theoretical results are supported by direct numerical simulation.

Chaotic transients, riddled basins, and stochastic transitions in coupled periodic logistic maps.

A system of two coupled map-based oscillators is studied. As units, we use identical logistic maps in two-periodic modes. In this system, increasing coupling strength significantly changes deterministic regimes of collective dynamics with coexisting periodic, quasiperiodic, and chaotic attractors. We study how random noise deforms these dynamical regimes in parameter zones of mono- and bistability, causes "order-chaos" transformations, and destroys regimes of in-phase and anti-phase synchronization. In the analytical study of these noise-induced phenomena, a stochastic sensitivity technique and a method of confidence domains for periodic and multi-band chaotic attractors are used. In this analysis, a key role of chaotic transients and geometry of "riddled" basins is revealed.

Stochastic transformations of multi-rhythmic dynamics and order-chaos transitions in a discrete 2D model.

A problem of the analysis of stochastic effects in multirhythmic nonlinear systems is investigated on the basis of the conceptual neuron map-based model proposed by Rulkov. A parameter zone with diverse scenarios of the coexistence of oscillatory regimes, both spiking and bursting, was revealed and studied. Noise-induced transitions between basins of periodic attractors are analyzed parametrically by statistics extracted from numerical simulations and by a theoretical approach using the stochastic sensitivity technique. Chaos-order transformations of dynamics caused by random forcing are discussed.

Canard oscillations in the randomly forced suspension flows.

Complex canard-type oscillatory regimes in stochastically forced flows of suspensions are studied. In this paper, we use the nonlinear dynamical model with a N-shaped rheological curve. Amplitude and frequency characteristics of self-oscillations in the zone of canard explosion are studied in dependence on the stiffness of this N-shaped function. A constructive role of random noise in the formation of complex oscillatory regimes is investigated. A phenomenon of the noise-induced splitting of stochastic cycles is discovered and studied both numerically and analytically by the stochastic sensitivity technique. Supersensitive canard cycles are described and their role in noise-induced transitions from order to chaos is discussed.

Noise-induced complex oscillatory dynamics in the Zeldovich-Semenov model of a continuous stirred tank reactor.

Noise-induced variability of thermochemical processes in a continuous stirred tank reactor is studied on the basis of the Zeldovich-Semenov dynamical model. For the deterministic variant of this model, mono- and bistability parametric zones as well as local and global bifurcations are determined. Noise-induced transitions between coexisting attractors (equilibria and cycles) and stochastic excitement with spike oscillations are investigated by direct numerical simulation and the analytical approach based on the stochastic sensitivity technique. For the stochastic model, the phenomenon of coherence resonance is discovered and studied.

Anomalous climate dynamics induced by multiplicative and additive noises.

Anomalous behavior of a nonlinear climate-vegetation model governed by the multiplicative and additive noises is revealed on the basis of stochastic sensitivity analysis. A specific feature of this model is the bistability with the coexistence of "snowball" equilibrium and "warm" attractor in the form of equilibrium or cycle. It is found that multiplicative and additive noises shift probabilistic distribution in opposite directions. The multiplicative noise introduced into the death rate of vegetation changes the dispersion of random states and their localization in the phase diagram. This type of noise cools down the system and is responsible for its transition to the snowball state. On the contrary, the additive noise warms up the climate with increasing noise intensity. A cumulative effect of multiplicative and additive noises occurs under their simultaneous influence. This effect determining the evolutionary behavior of a climate-vegetation system depends on the ratio of intensities of these noises.

Noise-induced spiking-bursting transition in the neuron model with the blue sky catastrophe.

We study a special variant of the noise-induced transition between spiking and bursting regimes associated with the blue sky catastrophe bifurcation in the Hindmarsh-Rose neuron model. We show that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, noise can transform the spiking oscillatory regime to the bursting one. This phenomenon is studied by means of power spectral density and interspike intervals statistics. We show that noise shifts the bifurcation value, so that bursting activity can be observed for a wider parameter range. Moreover, we reveal that the stochastic spiking-bursting transitions in this system are accompanied by the change in sign of the Lyapunov exponent. We perform a detailed quantitative analysis of these phenomena with an approach that uses a concept of the stochastic sensitivity function, the confidence domains method, and Mahalanobis metrics.

Noise-induced transitions and shifts in a climate-vegetation feedback model.

Motivated by the extremely important role of the Earth's vegetation dynamics in climate changes, we study the stochastic variability of a simple climate-vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate-vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate-vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth's vegetation cover is revealed. Our analysis demonstrates that the climate-vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.

Stochastic Sensitivity Analysis of Noise-Induced Extinction in the Ricker Model with Delay and Allee Effect.

A susceptibility of population systems to the random noise is studied on the base of the conceptual Ricker-type model taking into account the delay and Allee effect. This two-dimensional discrete model exhibits the persistence in the form of equilibria, discrete cycles, closed invariant curves, and chaotic attractors. It is shown how the Allee effect constrains the persistence zones with borders defined by crisis bifurcations. We study the role of random noise on the contraction and destruction of these zones. This phenomenon of the noise-induced extinction is investigated with the help of direct numerical simulations and semi-analytical approach based on the stochastic sensitivity functions. Stochastic transitions from the persistence regimes to the extinction are studied by the analysis of the mutual arrangement of the basins of attraction and confidence domains.

Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis.

A phenomenon of the noise-induced oscillatory multistability in glycolysis is studied. As a basic deterministic skeleton, we consider the two-dimensional Higgins model. The noise-induced generation of mixed-mode stochastic oscillations is studied in various parametric zones. Probabilistic mechanisms of the stochastic excitability of equilibria and noise-induced splitting of randomly forced cycles are analysed by the stochastic sensitivity function technique. A parametric zone of supersensitive Canard-type cycles is localized and studied in detail. It is shown that the generation of mixed-mode stochastic oscillations is accompanied by the noise-induced transitions from order to chaos.

Nonlinear dynamics of mushy layers induced by external stochastic fluctuations.

The time-dependent process of directional crystallization in the presence of a mushy layer is considered with allowance for arbitrary fluctuations in the atmospheric temperature and friction velocity. A nonlinear set of mushy layer equations and boundary conditions is solved analytically when the heat and mass fluxes at the boundary between the mushy layer and liquid phase are induced by turbulent motion in the liquid and, as a result, have the corresponding convective form. Namely, the 'solid phase-mushy layer' and 'mushy layer-liquid phase' phase transition boundaries as well as the solid fraction, temperature and concentration (salinity) distributions are found. If the atmospheric temperature and friction velocity are constant, the analytical solution takes a parametric form. In the more common case when they represent arbitrary functions of time, the analytical solution is given by means of the standard Cauchy problem. The deterministic and stochastic behaviour of the phase transition process is analysed on the basis of the obtained analytical solutions. In the case of stochastic fluctuations in the atmospheric temperature and friction velocity, the phase transition interfaces (mushy layer boundaries) move faster than in the deterministic case. A cumulative effect of these noise contributions is revealed as well. In other words, when the atmospheric temperature and friction velocity fluctuate simultaneously due to the influence of different external processes and phenomena, the phase transition boundaries move even faster. This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Stochastic sensitivity technique in a persistence analysis of randomly forced population systems with multiple trophic levels.

Motivated by important ecological applications we study how noise can reduce a number of trophic levels in hierarchically related multidimensional population systems. A nonlinear model with three trophic levels under the influence of external stochastic forcing is considered as a basic conceptual example. We analyze a probabilistic mechanism of noise-induced extinction of separate populations in this "prey-predator-top predator" system. We propose a new general mathematical approach for the estimation of the proximity of equilibrium regimes of this stochastic model to hazardous borders where abrupt changes in dynamics of ecological systems can occur. Our method is based on the stochastic sensitivity function technique and visualization method of confidence domains. Constructive abilities of this mathematical approach are demonstrated in the analysis of different scenaria of noise-induced reducing of the number of trophic levels.

How environmental noise can contract and destroy a persistence zone in population models with Allee effect.

A problem of the analysis of the noise-induced extinction in population models with Allee effect is considered. To clarify mechanisms of the extinction, we suggest a new technique combining an analysis of the geometry of attractors and their stochastic sensitivity. For the conceptual one-dimensional discrete Ricker-type model, on the base of the bifurcation analysis, deterministic persistence zones are constructed in the space of initial states and biological parameters. It is shown that the random environmental noise can contract, and even destroy these persistence zones. A parametric analysis of the probabilistic mechanism of the noise-induced extinction in regular and chaotic zones is carried out with the help of the unified approach based on the sensitivity functions technique and confidence domains method.

Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model.

We study the phenomenon of noise-induced torus bursting on the base of the three-dimensional Hindmarsh-Rose neuron model forced by additive noise. We show that in the parametric zone close to the Neimark-Sacker bifurcation, where the deterministic system exhibits rapid tonic spiking oscillations, random disturbances can turn tonic spiking into bursting, which is characterized by the formation of a peculiar dynamical structure resembling that of a torus. This phenomenon is confirmed by the changes in dispersion of random trajectories as well as the power spectral density and interspike intervals statistics. In particular, we show that as noise increases, the system undergoes P and D bifurcations, transitioning from order to chaos. We ultimately characterize the transition from stochastic (tonic) spiking to bursting by stochastic sensitivity functions.

Stochastic bifurcations caused by multiplicative noise in systems with hard excitement of auto-oscillations.

We study a stochastic dynamics of systems with hard excitement of auto-oscillations possessing a bistability mode with coexistence of the stable equilibrium and limit cycle. A principal difference in the results of the impact of additive and parametric random disturbances is shown. For the stochastic van der Pol oscillator with increasing parametric noise, qualitative transformations of the probability density function form "crater"-"peak+crater"-"peak" are demonstrated by numerical simulation. An analytical investigation of such P bifurcations is carried out for the stochastic Hopf-like model with hard excitement of self-oscillations. A detailed parametric description of the response of this model on the additive and multiplicative noise and corresponding stochastic bifurcations are presented and discussed.