Noise-induced Transitions Research Articles

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths for a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin–Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an n-dimensional system with a switching manifold in the drift that forms an (n−1)−dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use Γ−convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Emergency rate-driven control for rotor angle instability in power systems.

Renewable energy sources in modern power systems pose a serious challenge to the power system stability in the presence of stochastic fluctuations. Many efforts have been made to assess power system stability from the viewpoint of the bifurcation theory. However, these studies have not covered the dynamic evolution of renewable energy integrated, non-autonomous power systems. Here, we numerically explore the transition phenomena exhibited by a non-autonomous stochastic bi-stable power system oscillator model. We use additive white Gaussian noise to model the stochasticity in power systems. We observe that the delay in the transition observed for the variation of mechanical power as a function of time shows significant variations in the presence of noise. We identify that if the angular velocity approaches the noise floor before crossing the unstable manifold, the rate at which the parameter evolves has no control over the transition characteristics. In such cases, the response of the system is purely controlled by the noise, and the system undergoes noise-induced transitions to limit-cycle oscillations. Furthermore, we employ an emergency control strategy to maintain the stable non-oscillatory state once the system has crossed the quasi-static bifurcation point. We demonstrate an effective control strategy that opens a possibility of maintaining the stability of electric utility that operates near the physical limits.

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Abstract. We study the impact of applying stochastic forcing to the Ghil–Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and α-stable Lévy – α∈(0,2) – noise laws, examine the statistics of transition times, and estimate the most probable transition paths. While the Gaussian noise case – used here as a reference – has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, both in terms of mathematical theory and heuristics, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to −α, in apparent agreement with rigorous results obtained for additive noise in a related – yet different – reaction–diffusion equation and in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space, and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition paths cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

A probabilistic framework for windows of opportunity: the role of temporal variability in critical transitions.

The establishment of young organisms in harsh environments often requires a window of opportunity (WoO). That is, a short time window in which environmental conditions drop long enough below the hostile average level, giving the organism time to develop tolerance and transition into stable existence. It has been suggested that this kind of establishment dynamics is a noise-induced transition between two alternate states. Understanding how temporal variability (i.e. noise) in environmental conditions affects establishment of organisms is therefore key, yet not well understood or included explicitly in the WoO framework. In this paper, we develop a coherent theoretical framework for understanding when the WoO open or close based on simple dichotomous environmental variation. We reveal that understanding of the intrinsic timescales of both the developing organism and the environment is fundamental to predict if organisms can or cannot establish. These insights have allowed us to develop statistical laws for predicting establishment probabilities based on the period and variance of the fluctuations in naturally variable environments. Based on this framework, we now get a clear understanding of how changes in the timing and magnitude of climate variability or management can mediate establishment chances.

Two-stage noise-induced critical transitions in a fish population model with Allee effect in predators

Two-stage noise-induced critical transitions in a fish population model with Allee effect in predators

Transition and basin stability in a stochastic tumor growth model with immunization

The phenomenon of noise-induced transition driven by the correlated Gaussian and non-Gaussian colored noises is investigated by the first escape probability (FEP) and the mean first exit time (MFET). To derive the Markovian approximation of the original tumor growth model and obtain the analytical expressions of the FEP and MFET, we reduce the non-Gaussian colored noise and then expand the unified colored noise approximation (UCNA). Additionally, the stochastic basin of attraction (SBA), a recent geometric concept based on the FEP and MFET, is introduced to provide further insight into the effects of noisy fluctuations on the basin stability of a certain domain. A higher FEP or shorter MFET in the high tumor population region B facilitates the transition from B to the low tumor population region Bc, which indicates the weaker stability of domain B. Our main results demonstrate that (i) the transitions from B to Bc can be induced by both the Gaussian and non-Gaussian noise sources; (ii) the stronger noise intensity, especially the non-Gaussian noise intensity, with a larger deviation parameter and immune coefficient improves the FEP, shortens the MFET, and hence benefits the transitions. However, the enlargement of the correlation between noises strengthens the basin stability of domain B and impedes the transitions; (iii) the size of SBA expands due to the larger cross-correlated intensity. In contrast, the enhancements of noise intensities with a larger departure degree reduce the size of SBA, which weakens the basin stability and is less in favor of tumor treatment. Furthermore, the Monte Carlo simulations of the original system are employed to verify the feasibility and accuracy of the analytical predictions.

Tipping the Balance: A Criticality Perspective.

Cell populations are often characterised by phenotypic heterogeneity in the form of two distinct subpopulations. We consider a model of tumour cells consisting of two subpopulations: non-cancer promoting (NCP) and cancer-promoting (CP). Under steady state conditions, the model has similarities with a well-known model of population genetics which exhibits a purely noise-induced transition from unimodality to bimodality at a critical value of the noise intensity . The noise is associated with the parameter λ representing the system-environment coupling. In the case of the tumour model, λ has a natural interpretation in terms of the tissue microenvironment which has considerable influence on the phenotypic composition of the tumour. Oncogenic transformations give rise to considerable fluctuations in the parameter. We compute the phase diagram in a stochastic setting, drawing analogies between bifurcations and phase transitions. In the region of bimodality, a transition from a state of balance to a state of dominance, in terms of the competing subpopulations, occurs at λ = 0. Away from this point, the NCP (CP) subpopulation becomes dominant as λ changes towards positive (negative) values. The variance of the steady state probability density function as well as two entropic measures provide characteristic signatures at the transition point.

Stochastic sensitivity analysis and feedback control of noise-induced transitions in a predator-prey model with anti-predator behavior.

In this study, we investigate a stochastic predator-prey model with anti-predator behavior. We first analyze the noise-induced transition from a coexistence state to the prey-only equilibrium by using the stochastic sensitive function technique. The critical noise intensity for the occurrence of state switching is estimated by constructing confidence ellipses and confidence bands, respectively, for the coexistence the equilibrium and limit cycle. We then study how to suppress the noise-induced transition by using two different feedback control methods to stabilize the biomass at the attraction region of the coexistence equilibrium and the coexistence limit cycle, respectively. Our research indicates that compared with the prey population, the predators appear more vulnerable and prone to extinction in the presence of environmental noise, but it can be prevented by taking some appropriate feedback control strategies.

Noise-induced transition in the Zeldovich–Semenov model with local and global bifurcations

The noise-induced transition of the Zeldovich–Semenov model in a continuous stirred tank reactor is investigated under small random perturbations. The deterministic model will exhibit mono- and bistable characteristics via local and global bifurcations. In the bistable zone, based on the Freidlin–Wentzell large deviation theory, the stochastic preference is explained by analyzing the required action of the fluctuational path. For the case of monostability, in the weak noise limit, the emergence of the switching line gives rise to the sudden switch of the optimal path and the sliding cycle will appear via the sliding bifurcation, which is verified by numerical methods. In addition, when there is no saddle in phase space, stochastic excitation with large-amplitude spikes is studied. On the quasi-threshold manifold, the point with the minimum quasi-potential plays the same role as the saddle, which means that the optimal path will undergo a large excursion by crossing this special point. These phenomena are verified by employing stochastic simulations.

Existence of multiple noise-induced transitions in Lasota-Mackey maps.

We prove the existence of multiple noise-induced transitions in the Lasota-Mackey map, which is a class of one-dimensional random dynamical system with additive noise. The result is achieved with the help of rigorous computer assisted estimates. We first approximate the stationary distribution of the random dynamical system and then compute certified error intervals for the Lyapunov exponent. We find that the sign of the Lyapunov exponent changes at least three times when increasing the noise amplitude. We also show numerical evidence that the standard non-rigorous numerical approximation by finite-time Lyapunov exponent is valid with our model for a sufficiently large number of iterations. Our method is expected to work for a broad class of nonlinear stochastic phenomena.

Numerical study of noise-induced transitions in nonlinear dynamics of optically injected semiconductor lasers

Numerical study of noise-induced transitions in nonlinear dynamics of optically injected semiconductor lasers

A coral reef benthic system with grazing intensity and immigrated macroalgae in deterministic and stochastic environments

<abstract><p>As the most diverse, productive but vulnerable marine habitats in the world, coral reefs are easily affected by the ubiquitous environmental fluctuations, which could change the population dynamics and induce phase shifts of the ecosystem. In this paper, we consider a coral reef benthic system, where macroalgae, corals and algal turfs compete for the available space on a given region of the seabed with grazing intensity and immigrated macroalgae in both deterministic and stochastic environments. For the deterministic system, we analyze the existence and stability of equilibria, as well as the existence of bifurcations. For the stochastic system, sufficient conditions for the existence of the unique ergodic stationary distribution as well as the extinction of corals are obtained, by choosing suitable Lyapunov functions. Moreover, for the scenario that the system exhibits bistability between a macroalgal-coral coexistence equilibrium and a coral-free equilibrium in the absence of environmental fluctuation, we further investigate the irreversible noise-induced transition from macroalgal-coral coexistence to coral extirpation, and numerically estimate the critical values of noise intensity for the occurrence of such transition with the aids of the technique of stochastic sensitivity functions.</p></abstract>

Studying the Logistic Model

Several studies have used the logistic equation to model the growth of cancer cell populations 1 as seen in Eq. ( 1 ). This has included correlated multiplicative, [Formula: see text] and additive, [Formula: see text], noise terms. These noise terms can affect the growth rate, [Formula: see text], and death rate, [Formula: see text], of tumor cells and can be induced from factors such as radiotherapy or other cancer treatments. Depending on the intensity of the noise the terms, the fluctuations can induce a phase transition. Noise-induced transitions of nonlinear stochastic systems have applications in the fields of physics, chemistry and biology. [Formula: see text] We study the logistic differential equation with a multiplicative noise term before and at phase transition. Computational methods used to investigate this cancer cell model include a Diffusion Entropy Analysis method and a waiting time distribution method. 2 , 3 , 4 DEA will establish the scaling of a simulated series without altering the data through detrending. We hypothesize the treatment that causes a phase transition in the logistic model will induce tumor extinction and management. Understanding how to better evaluate and study cancer cell growth models will assist in assessing the efficacy of cancer treatments. Future work will include running simulations with a modified DEA method that includes the use of stripes. 2 For better statistics, the code will be adopted to run ensembles of simulated data instead of a single series. Generating and analyzing these large datasets can be computationally expensive. Through multiprocessing and the use of a supercomputer, we believe these computational limitations can be overcome.

Poisson-distributed noise induces cortical γ-activity: explanation of γ-enhancement by anaesthetics ketamine and propofol

Additive noise is known to affect the stability of nonlinear systems. To understand better the role of additive noise in neural systems, we investigate the impact of additive noise on a random neural network of excitatory and inhibitory neurons. Here we hypothesize that the noise originates from the ascending reticular activating system. Coherence resonance in the γ-frequency range emerges for intermediate noise levels while the network exhibits non-coherent activity at low and high noise levels. The analytical study of a corresponding mean-field model system explains the resonance effect by a noise-induced phase transition via a saddle-node bifurcation. An analytical study of the linear mean-field systems response to additive noise reveals that the coherent state exhibits a quasi-cycle in the γ-frequency range whose spectral properties are tuned by the additive noise. To illustrate the importance of the work, we show that the quasi-cycle explains γ-enhancement under impact of the anaesthetics ketamine and propofol as a destabilizing effect of the coherent state.

Multistability and noise-induced transitions in the model of bidirectionally coupled neurons with electrical synaptic plasticity

Multistability and noise-induced transitions in the model of bidirectionally coupled neurons with electrical synaptic plasticity

The role of noise in the tumor dynamics under chemotherapy treatment

Dynamical systems modeling tumor growth have been investigated to analyze the dynamics between tumor and healthy cells. Recent theoretical studies indicate that these interactions may lead to different dynamical outcomes under the effect of particular cancer therapies. In the present paper, we derive a system of nonlinear differential equations, in order to investigate solid tumors in vivo, taking into account the impact of chemotherapy on both tumor and healthy cells. We start by studying our model only in terms of deterministic dynamics under the variation of a drug concentration parameter. Later, with the introduction of noise, a stochastic model is used to analyze the impact of the unavoidable random fluctuations. As a result, new insights into noise-induced transitions are provided and illustrated in detail using techniques from dynamical systems and from the theory of stochastic processes.

A Stochastic Hierarchical Population System: Excitement, Extinction and Transition to Chaos

A problem of the mathematical modeling and analysis of noise-induced transformations of complex oscillatory regimes in hierarchical population systems is considered. As a key example, we use a three-dimensional food chain dynamical model of the interacting prey, predator, and top predator. We perform a comparative study of the impacts of random fluctuations on three key biological parameters of prey growth, predator mortality, and the top predator growth. A detailed investigation of the stochastic excitement, noise-induced transition from order to chaos, and various scenarios of extinction is carried out. Constructive abilities of the semi-analytical method of confidence domains in the analysis of the noise-induced extinction are demonstrated.

Stochastic sensitivity of Turing patterns: methods and applications to the analysis of noise-induced transitions

In this paper, a problem of the analysis of the randomly forced patterns in spatially distributed systems with diffusion is considered. For the approximation of mean-square deviations of random solutions from the unforced deterministic pattern-attractors, we suggest a constructive method based on the stochastic sensitivity technique. To demonstrate an efficiency of this method, we consider the Levin-Segel model with formation of non-homogeneous structures of the phytoplankton and herbivore populations. The spatial peculiarities of probabilistic distributions near patterns are investigated. The dependence of the stochastic sensitivity on the variation of system parameters is studied. An application of the stochastic sensitivity technique to the study of noise-induced transitions between coexisting spatial structures is demonstrated.

On Systems of Active Particles Perturbed by Symmetric Bounded Noises: A Multiscale Kinetic Approach

We consider an ensemble of active particles, i.e., of agents endowed by internal variables u(t). Namely, we assume that the nonlinear dynamics of u is perturbed by realistic bounded symmetric stochastic perturbations acting nonlinearly or linearly. In the absence of birth, death and interactions of the agents (BDIA) the system evolution is ruled by a multidimensional Hypo-Elliptical Fokker–Plank Equation (HEFPE). In presence of nonlocal BDIA, the resulting family of models is thus a Partial Integro-differential Equation with hypo-elliptical terms. In the numerical simulations we focus on a simple case where the unperturbed dynamics of the agents is of logistic type and the bounded perturbations are of the Doering–Cai–Lin noise or the Arctan bounded noise. We then find the evolution and the steady state of the HEFPE. The steady state density is, in some cases, multimodal due to noise-induced transitions. Then we assume the steady state density as the initial condition for the full system evolution. Namely we modeled the vital dynamics of the agents as logistic nonlocal, as it depends on the whole size of the population. Our simulations suggest that both the steady states density and the total population size strongly depends on the type of bounded noise. Phenomena as transitions to bimodality and to asymmetry also occur.

Stochastic variability of regular and chaotic dynamics in 2D metapopulation model

A behavior of metapopulation consisting of two coupled subsystems modeled by the Ricker map is considered. We study how dynamics of the metapopulation changes under increase in the intensity of migration between subpopulations. For the deterministic model, a variety of equilibrium, periodic, quasiperiodic, and chaotic attractors is described. An impact of random disturbances on the behavior of metapopulation is studied both numerically and analytically with the help of confidence domains. A phenomenon of the noise-induced temporal stabilization of the unstable equilibrium is discovered. We point out the special role of transients and fractal riddled basins in the noise-induced transitions from order to chaos.