Variability in the noise-induced modes of climate dynamics

New features of noise-induced climate variability are revealed on the basis of the three-dimensional model derived by Saltzman and Maasch. It is shown that the climate system can be highly noise excitable and it possesses the large-amplitude fluctuations even in those regions where its akin deterministic model does not contain any self-sustained oscillations. Intermittency in small- and large amplitude climate fluctuations between different basins of attraction of a limit cycle and stable equilibria substantially influencing the climate state (from warm to cold and vice versa) are found at various noise intensities. Suddenly occurring jumps between the basins of attraction of two stable equilibria corresponding to the warm and cold climate states are statistically confirmed under a certain diapason of noise intensities. The climate system undergoes transitions between its equilibria in the presence of noise in its prognostic variables. In addition, such transitions become more likely with increasing the noise intensity.     

Noise-induced reentrant transition of the stochastic Duffing oscillator

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise belongs to a well-defined interval. Noisy oscillations are found outside that range, i.e., for both weaker and stronger noise.Received: 20 February 2004, Published online: 20 April 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems     

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.     

Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given.     

Noise induced complexity: from subthreshold oscillations to spiking in coupled excitable systems

We study the stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. In the course of this transition diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed. In order to understand the details and mechanisms of these noise-induced dynamics we consider the thermodynamic limit N-->infinity of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good qualitative agreement with dynamical scenarios observed in the stochastic simulations of large ensembles.     

Statistical characteristics of pressure oscillations in a premixed combustor

This paper describes an investigation of the statistical characteristics of self-excited and noise-driven pressure oscillations in a premixed combustor. This work was motivated by observations that certain characteristics of these oscillations appear random and cannot be entirely characterized within a deterministic framework (e.g., spontaneous, noise-induced transitions of the combustor from stable to unstable operation or cycle-to-cycle variations in the oscillating pressure). In an effort to elucidate these stochastic elements, we performed an analysis of cycle-to-cycle variations in combustor pressure whose results are described in this paper. Data obtained from our combustor shows that the probability density function of the amplitude of these oscillations transitions from a Rayleigh to a Gaussian-type distribution as the combustor moves from stable to unstable operation. These data also show that the instability phase is nearly uniformly distributed; i.e., there is no phase value with maximum probability of occurrence. We also describe a theoretical analysis of the statistical features of a non-linear combustor model that is forced by random noise. Solutions of this model are presented and shown to be in agreement with measured data. The good agreement between the predictions and measured data suggest that the analysis presented in this paper provides a useful framework for interpreting many other apparently random features of combustor stability characteristics; for example, cyclic variability, “fuzziness” in stability boundaries, or noise-induced transitions.     

Noise-induced generation of saw-tooth type transitions between climate attractors and stochastic excitability of paleoclimate

Motivated by important paleoclimate applications we study a three dimensional model ofthe Quaternary climatic variations in the presence of stochastic forcing. It is shown thatthe deterministic system exhibits a limit cycle and two stable system equilibria. Wedemonstrate that the closer paleoclimate system to its bifurcation points (lying either inits monostable or bistable zone) the smaller noise generates small or large amplitudestochastic oscillations, respectively. In the bistable zone with two stable equilibria,noise induces a complex multimodal stochastic regime with intermittency of small and largeamplitude stochastic fluctuations. In the monostable zone, the small amplitude stochasticoscillations localized in the vicinity of unstable equilibrium appear along with the largeamplitude oscillations near the stable limit cycle. For the analysis of thesenoise-induced effects, we develop the stochastic sensitivity technique and use theMahalanobis metric in the three-dimensional case. To approximate the distribution ofrandom trajectories in Poincare sections, we use a method of confidence ellipses. Aspatial configuration of these ellipses is defined by the stochastic sensitivity and noiseintensity. The glaciation/deglaciation transitions going between two polar Earth’s stateswith the warm and cold climate become easier and quicker with increasing the noiseintensity. Our stochastic analysis demonstrates a near 100 ky saw-tooth type climate selffluctuations known from paleoclimate records. In addition, the enhancement of noiseintensity blurs the sharp climate cycles and reduces the glaciation-deglaciation periodsof the Earth’s paleoclimate.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Noise-induced chaos in non-linear dynamics of El Niños

Motivated by an important practical significance, we analyze the noise-induced El Niño evolutionary equations. Our analysis based on the evaluation of largest Lyapunov exponents demonstrates the new effects of deterministic and stochastic dynamics of the El Niño–Southern Oscillation Events. We show that the non-linear deterministic model possesses either a multiturn limit cycle with regular self-oscillations or chaotic oscillations depending on slight variations of one of the main system parameters – the mean tropical easterlies. It is revealed that in the presence of noise, transformations of regular oscillations into chaotic ones are observed.     

Random dynamics of the Morris-Lecar neural model

Determining the response characteristics of neurons to fluctuating noise-like inputs similar to realistic stimuli is essential for understanding neuronal coding. This study addresses this issue by providing a random dynamical system analysis of the Morris-Lecar neural model driven by a white Gaussian noise current. Depending on parameter selections, the deterministic Morris-Lecar model can be considered as a canonical prototype for widely encountered classes of neuronal membranes, referred to as class I and class II membranes. In both the transitions from excitable to oscillating regimes are associated with different bifurcation scenarios. This work examines how random perturbations affect these two bifurcation scenarios. It is first numerically shown that the Morris-Lecar model driven by white Gaussian noise current tends to have a unique stationary distribution in the phase space. Numerical evaluations also reveal quantitative and qualitative changes in this distribution in the vicinity of the bifurcations of the deterministic system. However, these changes notwithstanding, our numerical simulations show that the Lyapunov exponents of the system remain negative in these parameter regions, indicating that no dynamical stochastic bifurcations take place. Moreover, our numerical simulations confirm that, regardless of the asymptotic dynamics of the deterministic system, the random Morris-Lecar model stabilizes at a unique stationary stochastic process. In terms of random dynamical system theory, our analysis shows that additive noise destroys the above-mentioned bifurcation sequences that characterize class I and class II regimes in the Morris-Lecar model. The interpretation of this result in terms of neuronal coding is that, despite the differences in the deterministic dynamics of class I and class II membranes, their responses to noise-like stimuli present a reliable feature.     

Stochastic dynamics of the cubic map: A study of noise-induced transition phenomena

The effects of finite-amplitude, additive noise on the dynamics generated by a one-dimensional, two-parameter cubic map are considered. The underlying deterministic system exhibits bistability and hysteresis, and noise-induced processes associated with these phenomena are studied. If a bounded noise source is applied to this system, trajectories may be confined to a finite region. Mechanisms are given for the merging transitions between different parts of this region and the eventual escape from it as the noise level is increased. The noisy dynamics is also represented by an integral evolution operator, with an equilibrium density function with finite support. The operator's spectrum is determined as a function of map parameters and noise amplitude. Such noisy one-dimensional maps can provide models for the study of noise-induced phenomena described by stochastic differential equations.     

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.Dedicated to Ilya Prigogine on the occasion of his 70th birthday.     

Extreme events in the Ti:sapphire laser

We report experimental and theoretical evidence of the existence of extreme value events in the form of scarce and randomly emerging giant pulses in the femtosecond (self-pulsing or Kerr-lens mode-locked) Ti:sapphire laser. This laser displays complex dynamical behavior, including deterministic chaos, in two different regimes. The extreme value pulses are observed in the chaotic state of only one of these two regimes. The observations agree with the predictions of a well-tested theoretical model that does not include noise or self-Q-switching into its framework. This implies that, in this laser, the extreme effects have a nontrivial dynamical origin. The Ti:sapphire laser is hence revealed as a new and convenient system for the study of these effects.     

The effects of colored quadratic noise on a turbulent transition in liquid4He

Liquid⁴He presents an important physical system for the experimental study of noise-induced dynamical transitions. At temperatures belowT in the He II phase, the flow of heat in the liquid helium is limited by a kind of superfluid turbulence. The steady-state properties of this turbulence are adequately described by a dense tangle of quantized vortex lines in the superfluid component of the He II. The turbulence undergoes a continuous transition as the heat current is increased. At this transition the intrinsic fluctuations in the dissipation and the relaxation time both become large [D. Griswold, C. P. Lorenson, and J. T. Tough,Phys. Rev. B 35:3149 (1987)]. These observations are consistent with a model of the transition as an imperfect pitchfork bifurcation [M. Schumaker and W. Horsthemke,Phys. Rev. A 36:354 (1987)]. External noise can be easily added to the driving heat current. Small-amplitude noise simply causes the system to fluctuate about the deterministic steady states. Large-amplitude noise causes dramatic changes. The stochastic steady states of the turbulence show noise-induced bistability [D. Grisowld and J. T. Tough,Phys. Rev. A 36:1360 (1987)]. Comparison with the imperfect pitchfork model is difficult because the noise is colored, quadratic, and large. Nevertheless, an approximate result obtained by Schumaker and Horsthemke is in good qualitative agreement with the data.This paper will appear in a forthcoming issue of theJournal of Statistical Physics.     

Self-induced stochastic resonance in excitable systems

The effect of small-amplitude noise on excitable systems with strong time-scale separation is analyzed. It is found that vanishingly small random perturbations of the fast excitatory variable may result in the onset of a deterministic limit cycle behavior, absent without noise. The mechanism, termed self-induced stochastic resonance, combines a stochastic resonance-type phenomenon with an intrinsic mechanism of reset, and no periodic drive of the system is required. Self-induced stochastic resonance is different from other types of noise-induced coherent behaviors in that it arises away from bifurcation thresholds, in a parameter regime where the zero-noise (deterministic) dynamics does not display a limit cycle nor even its precursor. The period of the limit cycle created by the noise has a non-trivial dependence on the noise amplitude and the time-scale ratio between fast excitatory variables and slow recovery variables. It is argued that self-induced stochastic resonance may offer one possible scenario of how noise can robustly control the function of biological systems.     

Regular and chaotic dynamics of the dipole moment of square dipole arrays

Dipole lattices, which represent square dipole arrays, are investigated. Various types of equilibrium configurations of arrays are obtained, and conditions are shown under which these configurations are established. On the basis of parametric bifurcation diagrams, the main types of regular and chaotic oscillation regimes of the total dipole moment of a system are considered and their dependence on the amplitude, frequency, and polarization of an alternating field, as well as on the initial equilibrium configuration of arrays, is analyzed. Scenarios of the onset of chaotic regimes are demonstrated, including those that occur via the establishment and variation of quasiperiodic oscillations of the dipole moment of a system. The dynamic bistability state is revealed in which a stochastic resonance—an increase in the response of a system to a harmonic signal in the presence of noise—can be implemented.     

The nature of bursting noises,stochastic resonance and deterministic chaos in excitable neurons

We study the Hindmarsh-Rose model of excitable neurons and show that in the asymptotic limit this monostable model can possess some kind of dynamical bistability: small-amplitude quasiharmonic and large-amplitude relaxational oscillations can be simultaneously excited and their formation is accompanied by a narrow hysteresis. We show that bursting noises, stochastic resonance and deterministic chaos are determined by random transitions between these two dynamical states under slow and small changes of one of the model variables (z). We find that these effects take place even for such model parameters when hysteresis transforms into a step and they disappear when this step is smoothed out enough. We analyze some characteristics and conditions of formation of the deterministic chaos. We emphasize that such dynamical bistability and the effects related to it are universal phenomena and occur in a wide class of dynamical systems of different nature including brusselator.     

Additive noise in noise-induced nonequilibrium transitions

We study different nonlinear systems which possess noise-induced nonequlibrium transitions and shed light on the role of additive noise in these effects. We find that the influence of additive noise can be very nontrivial: it can induce first- and second-order phase transitions, can change properties of on-off intermittency, or stabilize oscillations. For the Swift-Hohenberg coupling, that is a paradigm in the study of pattern formation, we show that additive noise can cause the formation of ordered spatial patterns in distributed systems. We show also the effect of doubly stochastic resonance, which differs from stochastic resonance, because the influence of noise is twofold: multiplicative noise and coupling induce a bistability of a system, and additive noise changes a response of this noise-induced structure to the periodic driving. Despite the close similarity, we point out several important distinctions between conventional stochastic resonance and doubly stochastic resonance. Finally, we discuss open questions and possible experimental implementations. (c) 2001 American Institute of Physics.     

Supernarrow spectral peaks near a kinetic phase transition in a driven nonlinear micromechanical oscillator

We measure the spectral densities of fluctuations of an underdamped nonlinear micromechanical oscillator. By applying a sufficiently large periodic excitation, two stable dynamical states are obtained within a particular range of driving frequency. White noise is injected into the excitation, allowing the system to overcome the activation barrier and switch between the two states. While the oscillator predominately resides in one of the two states for most frequencies, a narrow range of frequencies exist where the occupations of the two states are approximately equal. At these frequencies, the oscillator undergoes a kinetic phase transition that resembles the phase transition of thermal equilibrium systems. We observe a supernarrow peak in the spectral densities of fluctuations of the oscillator. This peak is centered at the excitation frequency and arises as a result of noise-induced transitions between the two dynamical states.     

Nonequilibrium phase transitions induced by multiplicative noise: effects of self-correlation

A recently introduced lattice model, describing an extended system which exhibits a reentrant (symmetry-breaking, second-order) noise-induced nonequilibrium phase transition, is studied under the assumption that the multiplicative noise leading to the transition is colored. Within an effective Markovian approximation and a mean-field scheme it is found that when the self-correlation time tau of the noise is different from zero, the transition is also reentrant with respect to the spatial coupling D. In other words, at variance with what one expects for equilibrium phase transitions, a large enough value of D favors disorder. Moreover, except for a small region in the parameter subspace determined by the noise intensity sigma and D, an increase in tau usually prevents the formation of an ordered state. These effects are supported by numerical simulations.