Coherence resonance near a Hopf bifurcation

Trade-off between coherence and dissipation for excitable phase oscillators

Thermodynamic uncertainty relation (TUR) bounds coherence in stochastic oscillatory systems. In this paper, we show that both dynamical and thermodynamic bounds play important roles for the excitable oscillators, e.g., neurons. Firstly, we investigate the trade-off between coherence and dissipation both in the sub- and superthreshold regions for a single excitable unit, where both the TUR and the saddle-node on an invariant circle (SNIC) bounds constrain the fluctuation of interspike intervals. Secondly, we show that the widely studied phenomenon called coherence resonance, where there exists a noise strength to make the oscillatory responses of the system most coherent, is also bounded by the TUR in the one-dimensional excitable phase model. Finally, we study the coherence-dissipation relation in ensembles of strongly coupled excitable oscillators.

Fokker-Planck modeling of the stochastic dynamics of a Rijke tube

We derive and numerically validate a low-order oscillator model to capture the stochastic dynamics of a prototypical thermoacoustic system (a Rijke tube) undergoing a subcritical Hopf bifurcation in the presence of additive noise. We find that on the fixed-point branch before the bifurcation, the system is dominated by the first duct mode, and the Fokker-Planck solution for the first Galerkin mode can adequately predict the stochastic dynamics of the overall system. We also find that this analytical framework predicts well the dominant mode on the limit-cycle branch, but underperforms in the hysteretic bistable zone where the role of nonlinearities is more pronounced. Besides offering new insights into stochastic thermoacoustic behavior, this study shows that an analytical framework based on the Fokker-Planck equation can facilitate the early detection of thermoacoustic instabilities in a Rijke-tube model subjected to noise.

Effect of correlation time of combustion noise on early warning indicators of thermoacoustic instability

In this paper, we analyze the effects of finite correlation time (noise color) of combustion noise on noise-induced coherence and early warning indicators (EWIs) via numerical and experimental studies. We consider the Rijke tube as a prototypical combustion system and model combustion noise as an additive Ornstein-Uhlenbeck process while varying noise intensity and correlation time. We numerically investigate corresponding effects on coherence resonance and multi-fractal properties of pressure fluctuations. Subsequently, we experimentally validate results and elucidate the influence of noise color and intensity on trends in coherence resonance and multi-fractal measures that can be expected in a practical scenario using an electroacoustic simulator. We find that the coherence factor, which quantifies the relative contribution of coherent oscillations in a noisy signal, increases as the system approaches the thermoacoustic instability-irrespective of the correlation time. It works at most levels of combustion noise (except for too low and too high noise levels). The Hurst exponent reduces as the system approaches thermoacoustic instability only when the correlation time is small. These results have implications on the prediction and monitoring of thermoacoustic instability in practical combustors.

A universal description of stochastic oscillators

Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = μ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width μ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

Stochastic parametric excitation of convective heat transfer

We study the parametric excitation of the free thermal convection in a horizontal layer and a rectangular cell by random vertical vibrations. The mathematical formulation we use allows one to explore the cases of heating from below and above and the low-gravity conditions. The excitation threshold of the second moments of the current velocity and the temperature perturbations are derived. The heat flux through the system quantified by the Nusselt number is reported to be related to the second moment of temperature perturbations; therefore, the threshold of the stochastic excitation of second moments gives the threshold for the excitation of the convective heat transfer. Comparison of the stochastic parametric excitation with the effect of high-frequency periodic modulation reveals dramatic dissimilarity between the two. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Rijke tube: A nonlinear oscillator

Dynamical systems theory has emerged as an interdisciplinary area of research to characterize the complex dynamical transitions in real-world systems. Various nonlinear dynamical phenomena and bifurcations have been discovered over the decades using different reduced-order models of oscillators. Different measures and methodologies have been developed theoretically to detect, control, or suppress the nonlinear oscillations. However, obtaining such phenomena experimentally is often challenging, time-consuming, and risky mainly due to the limited control of certain parameters during experiments. With this review, we aim to introduce a paradigmatic and easily configurable Rijke tube oscillator to the dynamical systems community. The Rijke tube is commonly used by the combustion community as a prototype to investigate the detrimental phenomena of thermoacoustic instability. Recent investigations in such Rijke tubes have utilized various methodologies from dynamical systems theory to better understand the occurrence of thermoacoustic oscillations and their prediction and mitigation, both experimentally and theoretically. The existence of various dynamical behaviors has been reported in single and coupled Rijke tube oscillators. These behaviors include bifurcations, routes to chaos, noise-induced transitions, synchronization, and suppression of oscillations. Various early warning measures have been established to predict thermoacoustic instabilities. Therefore, this review article consolidates the usefulness of a Rijke tube oscillator in terms of experimentally discovering and modeling different nonlinear phenomena observed in physics, thus transcending the boundaries between the physics and the engineering communities.

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.

Interlayer Connectivity Affects the Coherence Resonance and Population Activity Patterns in Two-Layered Networks of Excitatory and Inhibitory Neurons

The firing patterns of neuronal populations often exhibit emergent collective oscillations, which can display substantial regularity even though the dynamics of individual elements is very stochastic. One of the many phenomena that is often studied in this context is coherence resonance, where additional noise leads to improved regularity of spiking activity in neurons. In this work, we investigate how the coherence resonance phenomenon manifests itself in populations of excitatory and inhibitory neurons. In our simulations, we use the coupled FitzHugh-Nagumo oscillators in the excitable regime and in the presence of neuronal noise. Formally, our model is based on the concept of a two-layered network, where one layer contains inhibitory neurons, the other excitatory neurons, and the interlayer connections represent heterotypic interactions. The neuronal activity is simulated in realistic coupling schemes in which neurons within each layer are connected with undirected connections, whereas neurons of different types are connected with directed interlayer connections. In this setting, we investigate how different neurophysiological determinants affect the coherence resonance. Specifically, we focus on the proportion of inhibitory neurons, the proportion of excitatory interlayer axons, and the architecture of interlayer connections between inhibitory and excitatory neurons. Our results reveal that the regularity of simulated neural activity can be increased by a stronger damping of the excitatory layer. This can be accomplished with a higher proportion of inhibitory neurons, a higher fraction of inhibitory interlayer axons, a stronger coupling between inhibitory axons, or by a heterogeneous configuration of interlayer connections. Our approach of modeling multilayered neuronal networks in combination with stochastic dynamics offers a novel perspective on how the neural architecture can affect neural information processing and provide possible applications in designing networks of artificial neural circuits to optimize their function via noise-induced phenomena.

White-noise-induced double coherence resonances in reduced Hodgkin-Huxley neuron model near subcritical Hopf bifurcation

Coherence resonance (CR) describes a counterintuitive phenomenon in which the optimal oscillatory responses in nonlinear systems are shaped by a suitable noise amplitude. This phenomenon has been observed in neural systems. In this research, the generation of double coherence resonances (DCRs) due to white noise is investigated in a three-dimensional reduced Hodgkin-Huxley neuron model with multiple-timescale feature. We show that additive white noise can induce DCRs from the resting state near a subcritical Hopf bifurcation. The appearance of DCRs is related to the changes of the firing pattern aroused by the increases of the noise amplitude. The underlying dynamical mechanisms for the appearance of the DCRs and the changes of the firing pattern are interpreted using the phase space analysis and the dynamics of the stable focus-node near the subcritical Hopf bifurcation. We find that the multiple-timescale dynamics is essential for generating the DCRs and different firing patterns. The results not only present a case in which noise can induce DCRs near a Hopf bifurcation but also provide its dynamical mechanism, which enriches the phenomena in nonlinear dynamics and provides further understanding on the roles of noise in neural systems with multiple-timescale feature.

Isostables for Stochastic Oscillators

Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.

Control of Timing Stability, and Suppression in Delayed Feedback Induced Frequency-Fluctuations by Means of Power Split Ratio and Delay Phase-Dependent Dual-Loop Optical Feedback

We experimentally demonstrated a power split ratio and optical delay phase dependent dual-loop optical feedback to investigate the suppression of frequency-fluctuations induced due to delayed optical feedback. The device under investigation is self-mode-locked (SML) two-section quantum-dash (QDash) laser operating at ∼21 GHz and emitting at ∼1.55 μm. The effect of two selective combinations of power split ratios (Loop-I: −23.29 dB and Loop-II: −28.06 dB, and Loop-I and Loop-II: −22 dB) and two optical delay phase settings ((i) stronger cavity set to integer resonance and fine-tuning the weaker cavity, (ii) weaker cavity set to integer resonance and fine-tuning of stronger cavity) on the suppression of cavity side-bands have been studied. Measured experimental results demonstrate that delayed optical feedback induced frequency-fluctuations can be effectively suppressed on integer resonance as well as on full delay range tuning (0–84 ps) by adjusting coupling strength −22 dB through Loop-I and Loop-II, respectively. Our findings suggest that power split ratio and delays phase-dependent dual-loop optical feedback can be used to maximize the performance of semiconductor mode-locked lasers.

Coherence resonance in neuronal populations: Mean-field versus network model

The counterintuitive phenomenon of coherence resonance describes a nonmonotonic behavior of the regularity of noise-induced oscillations in the excitable regime, leading to an optimal response in terms of regularity of the excited oscillations for an intermediate noise intensity. We study this phenomenon in populations of FitzHugh-Nagumo (FHN) neurons with different coupling architectures. For networks of FHN systems in an excitable regime, coherence resonance has been previously analyzed numerically. Here we focus on an analytical approach studying the mean-field limits of the globally and locally coupled populations. The mean-field limit refers to an averaged behavior of a complex network as the number of elements goes to infinity. We apply the mean-field approach to the globally coupled FHN network. Further, we derive a mean-field limit approximating the locally coupled FHN network with low noise intensities. We study the effects of the coupling strength and noise intensity on coherence resonance for both the network and the mean-field models. We compare the results of the mean-field and network frameworks and find good agreement in the globally coupled case, where the correspondence between the two approaches is sufficiently good to capture the emergence of coherence resonance, as well as of anticoherence resonance.

Coherence resonance in influencer networks

Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite the abundance of networks with influencers, little is known about their response to stochastic forcing. Here, for oscillatory dynamics on influencer networks, we show that subjecting influencers to an optimal intensity of noise can result in enhanced network synchronization. This new network dynamical effect, which we call coherence resonance in influencer networks, emerges from a synergy between network structure and stochasticity and is highly nonlinear, vanishing when the noise is too weak or too strong. Our results reveal that the influencer backbone can sharply increase the dynamical response in complex systems of coupled oscillators.

Acoustic vibrational resonance in a Rayleigh-Plesset bubble oscillator

The phenomenon of vibrational resonance (VR) has been investigated in a Rayleigh-Plesset oscillator for a gas bubble oscillating in an incompressible liquid while driven by a dual-frequency force consisting of high-frequency, amplitude-modulated, weak, acoustic waves. The complex equation of the Rayleigh-Plesset bubble oscillator model was expressed as the dynamics of a classical particle in a potential well of the Liénard type, thus allowing us to use both numerical and analytic approaches to investigate the occurrence of VR. We provide clear evidence that an acoustically-driven bubble oscillates in a time-dependent single or double-well potential whose properties are determined by the density of the liquid and its surface tension. We show both theoretically and numerically that, besides the VR effect facilitated by the variation of the parameters on which the high-frequency depends, amplitude modulation, the properties of the liquid in which the gas bubble oscillates contribute significantly to the occurrence of VR. In addition, we discuss the observation of multiple resonances and their origin for the double-well case, as well as their connection to the low frequency, weak, acoustic force field.

Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data

We present a framework for performing input-output system identification near a Hopf bifurcation using data from only the fixed-point branch, prior to the Hopf point itself. The framework models the system with a van der Pol-type equation perturbed by additive noise, and identifies the system parameters via the corresponding Fokker-Planck equation. We demonstrate the framework on a prototypical thermoacoustic oscillator (a flame-driven Rijke tube) undergoing a supercritical Hopf bifurcation. We find that the framework can accurately predict the properties of the Hopf bifurcation and the limit cycle beyond it. This study constitutes an experimental demonstration of system identification on a reacting flow using only prebifurcation data, opening up pathways to the development of early warning indicators for nonlinear dynamical systems near a Hopf bifurcation.

Phase descriptions of a multidimensional Ornstein-Uhlenbeck process

Stochastic oscillators play a prominent role in different fields of science. Their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case. One notion of phase that we put forward previously, the asymptotic phase of a stochastic oscillator, is based on the eigenfunction expansion of its probability density. More specifically, it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least-negative eigenvalue. Formally, besides the "backward" phase, one can also define the "forward" phase as the complex argument of the eigenfunction of the forward Kolomogorov operator corresponding to the least-negative eigenvalue. Until now, the intuition about these phase descriptions has been limited. Here we study these definitions for a process that is analytically tractable, the two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues. For this process, (i) we give explicit expressions for the two phases; (ii) we demonstrate that the isochrons are always the spokes of a wheel but that (iii) the spacing of these isochrons (their angular density) is different for backward and forward phases; (iv) we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field, whereas the forward phase also depends on the noise matrix; and (v) we demonstrate that the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case. We illustrate our analytical results for a number of qualitatively different cases.

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

Coherence resonance and stochastic bifurcation behaviors of simplified standing-wave thermoacoustic systems

In this work, noise-induced motions (i.e., external fluctuations) in two modelled standing-wave thermoacoustic systems are studied when these systems are close to the deterministic stability boundary. These systems include (1) open-open (i.e., Rijke-type) and (2) closed-open boundary conditions. It is found from the smooth transitions of the stationary probability density function that the thermoacoustic system is destabilized via stochastic P bifurcation, as the external noise intensity is continuously increased. In addition, the increased noise intensity can shift the hysteresis region, which makes the system more prone to quasi-periodic oscillations, but also reduces the hysteresis area. The noise-induced coherence motions are observed numerically in the open-open system, which is denoted by the occurrence of a bell-shaped signal to noise ratio (SNR). The SNR is shown to be applicable as a precursor. It becomes larger and the optimal noise intensity is decreased as the modelled thermoacoustic system approaches the critical bifurcation point. In addition, coherence resonance is observed in the closed-open system. To validate the findings, experimental studies are conducted on an open-open Rijke tube. Good qualitative agreements are obtained. The present study shed lights on the stochastic and coherence behaviors of the standing-wave thermoacoustic systems with different boundary conditions.

Noise-induced temporal regularity and signal amplification in an optomechanical system with parametric instability

Noise usually has an unwelcome influence on system performance. For instance, noise inevitably affects the low-frequency mechanical freedom in optomechanical experiments. However, we investigate here the beneficial effects of thermal noise on a basic optomechanical system with parametric instability. In a regime near parametric instability, it is found that thermal noise in the mechanical freedom can sustain long-term quasi-coherent oscillations when the system would otherwise remain in the equilibrium state. In an overlapping regime of parametric instability and bistability, intermittent switching between a self-sustained oscillating state and an equilibrium can be induced by adding a certain amount of noise. When a subthreshold periodic signal is applied to the mechanics, the switching between the self-sustained oscillations and the equilibrium exhibits good periodicity at a rate that is synchronized to the signal frequency, resulting in a significant amplification of the signal. Our results deepen the understanding of the interplay between optomechanical nonlinearity and noise and provide theoretical guidance for experimental observation of noise-induced beneficial phenomena in optomechanics.

Bistable emergence of oscillations in growing Bacillus subtilis biofilms

Biofilm communities of Bacillus subtilis bacteria have recently been shown to exhibit collective growth-rate oscillations mediated by electrochemical signaling to cope with nutrient starvation. These oscillations emerge once the colony reaches a large enough number of cells. However, it remains unclear whether the amplitude of the oscillations, and thus their effectiveness, builds up over time gradually or if they can emerge instantly with a nonzero amplitude. Here we address this question by combining microfluidics-based time-lapse microscopy experiments with a minimal theoretical description of the system in the form of a delay-differential equation model. Analytical and numerical methods reveal that oscillations arise through a subcritical Hopf bifurcation, which enables instant high-amplitude oscillations. Consequently, the model predicts a bistable regime where an oscillating and a nonoscillating attractor coexist in phase space. We experimentally validate this prediction by showing that oscillations can be triggered by perturbing the media conditions, provided the biofilm size lies within an appropriate range. The model also predicts that the minimum size at which oscillations start decreases with stress, a fact that we also verify experimentally. Taken together, our results show that collective oscillations in cell populations can emerge suddenly with nonzero amplitude via a discontinuous transition.

Multiple mechanisms for stochastic resonance are inherent to sinusoidally driven noisy Hopf oscillators

To ensure their sensitivity to weak periodic signals, some physical systems likely operate near a Hopf bifurcation. Many systems operating near such a bifurcation exhibit stochastic resonance, but it is unclear which mechanisms for resonance are inherent to the bifurcation. To address this question, we study the sinusoidally forced dynamics of noisy supercritical and subcritical Hopf oscillators. We find four qualitatively different mechanisms for stochastic resonance and determine the conditions for each type of resonance.

Bistability in two simple symmetrically coupled oscillators with symmetry-broken amplitude- and phase-locking

In the model system of two instantaneously and symmetrically coupled identical Stuart-Landau oscillators, we demonstrate that there exist stable solutions with symmetry-broken amplitude- and phase-locking. These states are characterized by a non-trivial fixed phase or amplitude relationship between both oscillators, while simultaneously maintaining perfectly harmonic oscillations of the same frequency. While some of the surrounding bifurcations have been previously described, we present the first detailed analytical and numerical description of these states and present analytically and numerically how they are embedded in the bifurcation structure of the system, arising both from the in-phase and the anti-phase solutions, as well as through a saddle-node bifurcation. The dependence of both the amplitude and the phase on parameters can be expressed explicitly with analytic formulas. As opposed to the previous reports, we find that these symmetry-broken states are stable, which can even be shown analytically. As an example of symmetry-breaking solutions in a simple and symmetric system, these states have potential applications as bistable states for switches in a wide array of coupled oscillatory systems.

Weak multiplexing induces coherence resonance

Using the model of a FitzHugh-Nagumo system in the excitable regime, we study the impact of multiplexing on coherence resonance in a two-layer network. We show that multiplexing allows for the control of the noise-induced dynamics. In particular, we find that multiplexing induces coherence resonance in networks that do not demonstrate this phenomenon in isolation. Examples are provided by deterministic networks and networks where the strength of interaction between the elements is not optimal for coherence resonance. In both cases, we show that the control strategy based on multiplexing can be successfully applied even for weak coupling between the layers. Moreover, for the case of deterministic networks, we obtain a counter-intuitive result: the multiplex-induced coherence resonance in the layer which is deterministic in isolation manifests itself even more strongly than that in the noisy layer.

Time-delayed feedback control of coherence resonance chimeras

Using the model of a FitzHugh-Nagumo system in the excitable regime, we investigate the influence of time-delayed feedback on noise-induced chimera states in a network with nonlocal coupling, i.e., coherence resonance chimeras. It is shown that time-delayed feedback allows for the control of the range of parameter values where these chimera states occur. Moreover, for the feedback delay close to the intrinsic period of the system, we find a novel regime which we call period-two coherence resonance chimera.

Vibrational resonance in an inhomogeneous medium with periodic dissipation

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Heat dissipation and information flow for delayed bistable Langevin systems near coherence resonance

In this paper, stochastic thermodynamics of delayed bistable Langevin systems near coherence resonance is discussed. We calculate the heat dissipation rate and the information flow of a delayed bistable Langevin system under various noise intensities. Both the heat dissipation rate and the information flow are found to be bell-shaped functions of the noise intensity, which implies that coherence resonance manifests itself in the thermodynamic properties.

Stochastic ice stream dynamics

Ice streams are narrow corridors of fast-flowing ice that constitute the arterial drainage network of ice sheets. Therefore, changes in ice stream flow are key to understanding paleoclimate, sea level changes, and rapid disintegration of ice sheets during deglaciation. The dynamics of ice flow are tightly coupled to the climate system through atmospheric temperature and snow recharge, which are known exhibit stochastic variability. Here we focus on the interplay between stochastic climate forcing and ice stream temporal dynamics. Our work demonstrates that realistic climate fluctuations are able to (i) induce the coexistence of dynamic behaviors that would be incompatible in a purely deterministic system and (ii) drive ice stream flow away from the regime expected in a steady climate. We conclude that environmental noise appears to be crucial to interpreting the past behavior of ice sheets, as well as to predicting their future evolution.

Stochastic bifurcations in a prototypical thermoacoustic system

We study the influence of noise in a prototypical thermoacoustic system, which represents a nonlinear self-excited bistable oscillator. We analyze the time series of unsteady pressure obtained from a horizontal Rijke tube and a mathematical model to identify the effect of noise. We report the occurrence of stochastic bifurcations in a thermoacoustic system by tracking the changes in the stationary amplitude distribution. We observe a complete suppression of a bistable zone in the presence of high intensity noise. We find that the complete suppression of the bistable zone corresponds to the nonexistence of phenomenological (P) bifurcations. This is a study in thermoacoustics to identify the parameter regimes pertinent to P bifurcation using the stationary amplitude distribution obtained by solving the Fokker-Planck equation.

Coherence-Resonance Chimeras in a Network of Excitable Elements

We demonstrate that chimera behavior can be observed in nonlocally coupled networks of excitable systems in the presence of noise. This phenomenon is distinct from classical chimeras, which occur in deterministic oscillatory systems, and it combines temporal features of coherence resonance, i.e., the constructive role of noise, and spatial properties of chimera states, i.e., the coexistence of spatially coherent and incoherent domains in a network of identical elements. Coherence-resonance chimeras are associated with alternating switching of the location of coherent and incoherent domains, which might be relevant in neuronal networks.

Coherence resonance in a thermoacoustic system

We experimentally investigated the noise-induced dynamics of a prototypical thermoacoustic system undergoing a subcritical Hopf bifurcation to limit cycle oscillations. The study was performed prior to the bistable regime. Analysis of the characteristics of pressure oscillations in the combustor and fluctuations in the heat release rate from the flame-the two physical entities involved in thermoacoustic coupling-at increasing levels of noise indicated precursors to the Hopf bifurcation. These precursors were further identified to be a result of coherence resonance.

Coherence enhanced intermittency in an optically injected semiconductor laser

We report on the experimental observation of coherence enhancement of noise-induced intermittency in a semiconductor laser subject to optical injection from another laser at the boundary of the frequency-locking regime. The intermittent switches between locked and unlocked states occur more regularly at a certain value of the injecting laser pump current. A shape of probability distribution of the experimental inter-spike-interval fluctuations is used to quantitatively characterize the intermittent behavior.

Intrinsic noise induced coherence resonance in a glow discharge plasma

Experimental evidence of intrinsic noise induced coherence resonance in a glow discharge plasma is being reported. Initially the system is started at a discharge voltage (DV) where it exhibited fixed point dynamics, and then with the subsequent increase in the DV spikes were excited which were few in number and with further increase of DV the number of spikes as well as their regularity increased. The regularity in the interspike interval of the spikes is estimated using normalized variance. Coherence resonance was determined using normalized variance curve and also corroborated by Hurst exponent and power spectrum plots. We show that the regularity of the excitable spikes in the floating potential fluctuation increases with the increase in the DV, up to a particular value of DV. Using a Wiener filter, we separated the noise component which was observed to increase with DV and hence conjectured that noise can play an important role in the generation of the coherence resonance. From an anharmonic oscillator equation describing ion acoustic oscillations, we have been able to obtain a FitzHugh-Nagumo like model which has been used to understand the excitable dynamics of glow discharge plasma in the presence of noise. The numerical results agree quite well with the experimental results.

Time-delayed feedback control of coherence resonance near subcritical Hopf bifurcation: theory versus experiment

Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation, we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay, either suppression or enhancement of coherence resonance can be achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.

Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping

Excitability and coherence resonance are studied in a semiconductor quantum dot laser under short optical self-feedback. For low pump levels, these are observed close to a homoclinic bifurcation, which is in correspondence with earlier observations in quantum well lasers. However, for high pump levels, we find excitability close to a boundary crisis of a chaotic attractor. We demonstrate that in contrast to the homoclinic bifurcation the crisis and thus the excitable regime is highly sensitive to the pump current. The excitability threshold increases with the pump current, which permits to adjust the sensitivity of the excitable unit to noise as well as to shift the optimal noise strength, at which maximum coherence is observed. The shift adds up to more than one order of magnitude, which strongly facilitates experimental realizations.

Event-triggered feedback in noise-driven phase oscillators

Using a stochastic nonlinear phase oscillator model, we study the effect of event-triggered feedback on the statistics of interevent intervals. Events are associated with the entering of a new cycle. The feedback is modeled by an instantaneous increase (positive feedback) or decrease (negative feedback) of the oscillator frequency whenever an event occurs followed by an exponential decay on a slow time scale. In addition to the known excitable and oscillatory regimes, which are separated by a saddle node on invariant circle bifurcation, positive feedback can lead to bistable dynamics and a change of the system's excitability. The feedback has also a strong effect on noise-induced phenomena like coherence resonance or anticoherence resonance. Both positive and negative feedback can lead to more regular output for particular noise strengths. Finally, we investigate serial correlations in the sequence of interevent intervals that occur due to the additional slow dynamics. We derive approximations for the serial correlation coefficient and show that positive feedback results in extended positive interval correlations, whereas negative feedback yields short-ranging negative correlations. Investigating the interplay of feedback and the nonlinear phase dynamics close to the bifurcation, we find that correlations are most pronounced for optimal feedback strengths.

Temperature-driven coherence resonance and stochastic resonance in a thermochemical system

We perform the stochastic analysis of a thermochemical system using a master equation which describes a chemical reaction and includes discrete and continuous temperature jumps. We study the time evolution of the system selecting the temperature of the thermostat as an easily tunable control parameter. Depending on the thermostat temperature, the system can be in an excitable, oscillatory, or stationary regime. Stochastic time series for the system temperature are generated and the distributions of interspike intervals are analyzed in the three dynamical regimes separated by a homoclinic bifurcation and a Hopf bifurcation. Different constructive roles of internal fluctuations are exhibited. A noise-induced transition is observed in the vicinity of the Hopf bifurcation. Coherence resonance and stochastic resonance are found in the oscillatory regime. In a range of thermostat temperatures, a nontrivial behavior of the highly nonlinear system is revealed by the existence of both a minimum and a maximum in the scaled standard deviation of interspike intervals as a function of particle number. This high sensitivity to system size illustrates that controlling dynamics in nanoreactors may remain a difficult task.

Stochastic sensitivity analysis of the noise-induced excitability in a model of a hair bundle

We study effect of weak noise on the dynamics of a hair bundle model near the excitability threshold and near a subcritical Hopf bifurcation. We analyze numerically noise-induced structural changes in the probability density and the power spectral density of the model. In particular, we show that weak noise can induce oscillations with two distinct frequencies in both excitable and limit-cycle regimes. We then applied a recently developed technique of stochastic sensitivity functions which allows us to estimate threshold values of noise intensity corresponding to these transitions.

A mean-field approach to elastically coupled hair bundles

We study the dynamics of oscillatory hair bundles which are coupled elastically in their deflection variable and are subject to noise. We present a stochastic description capturing the dynamics of the hair bundles' mean field. In particular, the presented derivation elucidates the origin of the previously described noise reduction by coupling. By comparison of simulations of the approximate dynamics and the full system, we verify our results. Furthermore, we demonstrate that the specific type of coupling considered implies coupling-induced changes in the dynamics beyond mere noise reduction.

Analysing dynamical behavior of cellular networks via stochastic bifurcations

The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation. However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood. Here, we define a concept of stochastic bifurcations suitable to investigate the dynamical structure of genetic networks, and show that under stochastic influence, the expression of given proteins of interest is defined via the probability distribution of the phase variable, representing one of the genes constituting the system. Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types.

Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise

This paper aims to investigate Gaussian colored-noise-induced stochastic bifurcations and the dynamical influence of correlation time and noise intensity in a bistable Duffing-Van der Pol oscillator. By using the stochastic averaging method, theoretically, one can obtain the stationary probability density function of amplitude for the Duffing-Van der Pol oscillator and can reveal interesting dynamics under the influence of Gaussian colored noise. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that system parameters, noise intensity, and noise correlation time, respectively, can be treated as bifurcation parameters. They also imply that the effects of multiplicative noise are rather different from that of additive noise. The results of direct numerical simulation verify the effectiveness of the theoretical analysis. Moreover, the largest Lyapunov exponent calculations indicate that P and D bifurcations have no direct connection.

Architecture-dependent robustness and bistability in a class of genetic circuits

Understanding the relationship between genotype and phenotype is a challenge in systems biology. An interesting yet related issue is why a particular circuit topology is present in a cell when the same function can supposedly be obtained from an alternative architecture. Here we analyzed two topologically equivalent genetic circuits of coupled positive and negative feedback loops, named NAT and ALT circuits, respectively. The computational search for the oscillation volume of the entire biologically reasonable parameter region through large-scale random samplings shows that the NAT circuit exhibits a distinctly larger fraction of the oscillatory region than the ALT circuit. Such a global robustness difference between two circuits is supplemented by analyzing local robustness, including robustness to parameter perturbations and to molecular noise. In addition, detailed dynamical analysis shows that the molecular noise of both circuits can induce transient switching of the different mechanism between a stable steady state and a stable limit cycle. Our investigation on robustness and dynamics through examples provides insights into the relationship between network architecture and its function.

Coherence resonance due to transient thresholds in excitable systems

Excitable systems can have more than one response threshold, but accessing each of these is only facilitated by preferential choice of the appropriate components in the input noise. The coherence resonance phenomenon discovered by Pikovsky and Kurths [Phys. Rev. Lett. 78, 775 (1997)] utilizes only one response threshold, thus leaving the nature of the dynamics of a possible second threshold unspecified. Here we show using a FitzHugh-Nagumo excitable system that the second response threshold can be reached transiently by brief pulses in the negative noise component, leading to a coherence resonance phenomenon of its own. The resonance can occur both as a function of input amplitude and frequency. The phenomenon is also illustrated in more realistic Hodgkin-Huxley model equations, and analytical predictions are made using probabilistic considerations of the input. This phenomenon attributes more complex role noise can play in excitable systems.

Dynamics of two semiconductor lasers coupled by a passive resonator

The stability of two semiconductor lasers that are spatially separated by a passive resonator is analyzed using the composite-cavity mode approach. We study the nonlinear interactions of three composite-cavity modes and identify regions of in-phase and out-of-phase laser locking in the parameter plane of the transmission coefficients of the coupling mirrors and the laser length difference. Bifurcation analysis shows that the structure of the locking regions strongly depends on (i) the length of the passive resonator and (ii) the amount of amplitude-phase coupling of the laser field. Specifically, we find a single locking region when the passive resonator and the lasers have comparable lengths and up to three separate locking regions when the passive resonator is much shorter than the lasers. Furthermore, we use the recently developed 0-1 test for chaos to uncover intricate regions of chaotic dynamics that shrink in size and eventually disappear as the passive resonator length becomes comparable to the laser length.

Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator

We investigate the influence of additive Gaussian white noise on two different bistable self-sustained oscillators: Duffing-Van der Pol oscillator with hard excitation and a model of a synthetic genetic oscillator. In the deterministic case, both oscillators are characterized with a coexistence of a stable limit cycle and a stable equilibrium state. We find that under the influence of noise, their dynamics can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary amplitude distribution. For the Duffing-Van der Pol oscillator analytical results, obtained for a quasiharmonic approach, are compared with the result of direct computer simulations. In particular, we show that the dynamics is different for isochronous and anisochronous systems. Moreover, we find that the increase of noise intensity in the isochronous regime leads to a narrowing of the spectral line. This effect is similar to coherence resonance. However, in the case of anisochronous systems, this effect breaks down and a new phenomenon, anisochronous-based stochastic bifurcation occurs.

Spontaneous movements and linear response of a noisy oscillator

A deterministic system that operates in the vicinity of a Hopf bifurcation can be described by a single equation of a complex variable, called the normal form. Proximity to the bifurcation ensures that on the stable side of the bifurcation (i.e. on the side where a stable fixed point exists), the linear-response function of the system is peaked at the frequency that is characteristic of the oscillatory instability. Fluctuations, which are present in many systems, conceal the Hopf bifurcation and lead to noisy oscillations. Spontaneous hair bundle oscillations by sensory hair cells from the vertebrate ear provide an instructive example of such noisy oscillations. By starting from a simplified description of hair bundle motility based on two degrees of freedom, we discuss the interplay of nonlinearity and noise in the supercritical Hopf normal form. Specifically, we show here that the linear-response function obeys the same functional form as for the noiseless system on the stable side of the bifurcation but with effective, renormalized parameters. Moreover, we demonstrate in specific cases how to relate analytically the parameters of the normal form with added noise to effective parameters. The latter parameters can be measured experimentally in the power spectrum of spontaneous activity and linear-response function to external stimuli. In other cases, numerical solutions were used to determine the effects of noise and nonlinearities on these effective parameters. Finally, we relate our results to experimentally observed spontaneous hair bundle oscillations and responses to periodic stimuli.

Resonant control of stochastic spatiotemporal dynamics in a tunnel diode by multiple time-delayed feedback

We study the control of noise-induced spatiotemporal current density patterns in a semiconductor nanostructure (double-barrier resonant tunneling diode) by multiple time-delayed feedback. We find much more pronounced resonant features of noise-induced oscillations compared to single time feedback, rendering the system more sensitive to variations in the delay time tau . The coherence of noise-induced oscillations measured by the correlation time exhibits sharp resonances as a function of tau , and can be strongly increased by optimal choices of tau . Similarly, the peaks in the power spectral density are sharpened. We provide analytical insight into the control mechanism by relating the correlation times and mean frequencies of noise-induced breathing oscillations to the stability properties of the deterministic stationary current density filaments under the influence of the control loop. Moreover, we demonstrate that the use of multiple time delays enlarges the regime in which the deterministic dynamical properties of the system are not changed by delay-induced bifurcations.

Control of coherence resonance in semiconductor superlattices

We study the effect of time-delayed feedback control and Gaussian white noise on the spatiotemporal charge dynamics in a semiconductor superlattice. The system is prepared in a regime where the deterministic dynamics is close to a global bifurcation, namely, a saddle-node bifurcation on a limit cycle. In the absence of control, noise can induce electron charge front motion through the entire device, and coherence resonance is observed. We show that with appropriate selection of the time-delayed feedback parameters the effect of coherence resonance can be either enhanced or destroyed, and the coherence of stochastic domain motion at low noise intensity is dramatically increased. Additionally, the purely delay-induced dynamics in the system is investigated, and a homoclinic bifurcation of a limit cycle is found.

Roles of external noise correlation in optimal intracellular calcium signaling

The dynamics of a minimal calcium model, which is subjected to white noise or colored noise, was investigated. For white noise, coherence of noise-induced calcium oscillations reached a maximum at an optimal noise intensity, characterizing coherence resonance. Higher resonance peaks could be observed at lower noise intensity when a control parameter is tuned to approach a bifurcation point. For colored noise, a maximal coherence of the oscillations was found for suitable values of both the intensity and the correlation time. Moreover, the coherence of the oscillations exhibited two maxima at two values of noise intensity (correlation time) for appropriate noise correlation time (intensity). In addition, a quantitative description of the effects of noise correlation time on the resonance behavior was presented. The resonance behavior, which is induced either by white noise or colored noise, was interpreted by terms of height and relative width of a spectral peak.