Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

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.

Tipping point and noise-induced transients in ecological networks

A challenging and outstanding problem in interdisciplinary research is to understand the interplay between transients and stochasticity in high-dimensional dynamical systems. Focusing on the tipping-point dynamics in complex mutualistic networks in ecology constructed from empirical data, we investigate the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are studied: environmental (Gaussian white) noise and state-dependent demographic noise. The dynamical mechanism responsible for both phenomena is a transition from one stable steady state to another driven by stochastic forcing, mediated by an unstable steady state. Exploiting a generic and effective two-dimensional reduced model for real-world mutualistic networks, we find that the average transient lifetime scales algebraically with the noise amplitude, for both environmental and demographic noise. We develop a physical understanding of the scaling laws through an analysis of the mean first passage time from one steady state to another. The phenomena of noise-induced collapse and recovery and the associated scaling laws have implications for managing high-dimensional ecological systems.

Multistability, chaos, and random signal generation in semiconductor superlattices

Historically, semiconductor superlattices, artificial periodic structures of different semiconductor materials, were invented with the purpose of engineering or manipulating the electronic properties of semiconductor devices. A key application lies in generating radiation sources, amplifiers, and detectors in the "unusual" spectral range of subterahertz and terahertz (0.1-10 THz), which cannot be readily realized using conventional radiation sources, the so-called THz gap. Efforts in the past three decades have demonstrated various nonlinear dynamical behaviors including chaos, suggesting the potential to exploit chaos in semiconductor superlattices as random signal sources (e.g., random number generators) in the THz frequency range. We consider a realistic model of hot electrons in semiconductor superlattice, taking into account the induced space charge field. Through a systematic exploration of the phase space we find that, when the system is subject to an external electrical driving of a single frequency, chaos is typically associated with the occurrence of multistability. That is, for a given parameter setting, while there are initial conditions that lead to chaotic trajectories, simultaneously there are other initial conditions that lead to regular motions. Transition to multistability, i.e., the emergence of multistability with chaos as a system parameter passes through a critical point, is found and argued to be abrupt. Multistability thus presents an obstacle to utilizing the superlattice system as a reliable and robust random signal source. However, we demonstrate that, when an additional driving field of incommensurate frequency is applied, multistability can be eliminated, with chaos representing the only possible asymptotic behavior of the system. In such a case, a random initial condition will lead to a trajectory landing in a chaotic attractor with probability 1, making quasiperiodically driven semiconductor superlattices potentially as a reliable device for random signal generation to fill the THz gap. The interplay among noise, multistability, and chaos is also investigated.

How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.

The joy of transient chaos

We intend to show that transient chaos is a very appealing, but still not widely appreciated, subfield of nonlinear dynamics. Besides flashing its basic properties and giving a brief overview of the many applications, a few recent transient-chaos-related subjects are introduced in some detail. These include the dynamics of decision making, dispersion, and sedimentation of volcanic ash, doubly transient chaos of undriven autonomous mechanical systems, and a dynamical systems approach to energy absorption or explosion.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system.

A heuristic method for identifying chaos from frequency content

The sign of the largest Lyapunov exponent is the fundamental indicator of chaos in a dynamical system. However, although the extraction of Lyapunov exponents can be accomplished with (necessarily noisy) the experimental data, this is still a relatively data-intensive and sensitive endeavor. This paper presents an alternative pragmatic approach to identifying chaos using response frequency characteristics and extending the concept of the spectrogram. The method is shown to work well on both experimental and simulated time series.

Fractal snapshot components in chaos induced by strong noise

In systems exhibiting transient chaos in coexistence with periodic attractors, the inclusion of weak noise might give rise to noise-induced chaotic attractors. When the noise amplitude exceeds a critical value, an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set. A further increase of noise leads to a fuzzy nonfractal pattern. By means of the concept of snapshot attractors and random maps, we point out that the fuzzy pattern can be decomposed into well-defined fractal components, the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories. The pattern of the snapshot attractor and its characteristic numbers, such as the finite time Lyapunov exponents and numerically evaluated fractal dimensions, change continuously in time. We find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size. The validity of the Kaplan-Yorke formula is also investigated. A superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise-induced attractor at the same noise strength.

Irregularly appearing early afterdepolarizations in cardiac myocytes: random fluctuations or dynamical chaos?

Irregularly occurring early afterdepolarizations (EADs) in cardiac myocytes are traditionally hypothesized to be caused by random ion channel fluctuations. In this study, we combined 1), patch-clamp experiments in which action potentials were recorded at different pacing cycle lengths from isolated rabbit ventricular myocytes under several experimental conditions inducing EADs, including oxidative stress with hydrogen peroxide, calcium overload with BayK8644, and ionic stress with hypokalemia; 2), computer simulations using a physiologically detailed rabbit ventricular action potential model, in which repolarization reserve was reduced to generate EADs and random ion channel or path cycle length fluctuations were implemented; and 3), iterated maps with or without noise. By comparing experimental, modeling, and bifurcation analyses, we present evidence that noise-induced transitions between bistable states (i.e., between an action potential with and without an EAD) is not sufficient to account for the large variation in action potential duration fluctuations observed in experimental studies. We conclude that the irregular dynamics of EADs is intrinsically chaotic, with random fluctuations playing a nonessential, auxiliary role potentiating the complex dynamics.

Quasipotential approach to critical scaling in noise-induced chaos

When a dynamical system exhibits transient chaos and a nonchaotic attractor, as in a periodic window, noise can induce a chaotic attractor. In particular, when the noise amplitude exceeds a critical value, the largest Lyapunov exponent of the attractor of the system starts to increase from zero. While a scaling law for the variation of the Lyapunov exponent with noise was uncovered previously, it is mostly based on numerical evidence and a heuristic analysis. This paper presents a more general approach to the scaling law, one based on the concept of quasipotentials. Besides providing deeper insights into the problem of noise-induced chaos, the quasipotential approach enables previously unresolved issues to be addressed. The fractal properties of noise-induced chaotic attractors and applications to biological systems are also discussed.

Chaotic Red Queen coevolution in three-species food chains

Coevolution between two antagonistic species follows the so-called 'Red Queen dynamics' when reciprocal selection results in an endless series of adaptation by one species and counteradaptation by the other. Red Queen dynamics are 'genetically driven' when selective sweeps involving new beneficial mutations result in perpetual oscillations of the coevolving traits on the slow evolutionary time scale. Mathematical models have shown that a prey and a predator can coevolve along a genetically driven Red Queen cycle. We found that embedding the prey-predator interaction into a three-species food chain that includes a coevolving superpredator often turns the genetically driven Red Queen cycle into chaos. A key condition is that the prey evolves fast enough. Red Queen chaos implies that the direction and strength of selection are intrinsically unpredictable beyond a short evolutionary time, with greatest evolutionary unpredictability in the superpredator. We hypothesize that genetically driven Red Queen chaos could explain why many natural populations are poised at the edge of ecological chaos. Over space, genetically driven chaos is expected to cause the evolutionary divergence of local populations, even under homogenizing environmental fluctuations, and thus to promote genetic diversity among ecological communities over long evolutionary time.

Stochastic resonance in a generalized quantum Kubo oscillator

We discuss stochastic resonance in a biased linear quantum system that is subject to multiplicative and additive noises. Starting from a microscopic system-reservoir Hamiltonian, we derive a c-number analogue of the generalized Langevin equation. The developed approach puts forth a quantum mechanical generalization of the "Kubo type" oscillator which is a linear system. Such a system is often used in the literature to study various phenomena in nonequilibrium systems via a particular interaction between system and the external noise. Our analytical results proposed here have the ability to reveal the role of external noise and vis-a-vis the mechanisms and detection of subtle underlying signatures of the stochastic resonance behavior in a linear system. In our development, we show that only when the external noise possesses a "finite correlation time" the quantum effect begins to appear. We observe that the quantum effect enhances the resonance in comparison to the classical one.

Onset of colored-noise-induced synchronization in chaotic systems

We develop and validate an algorithm for integrating stochastic differential equations under green noise. Utilizing it and the standard methods for computing dynamical systems under red and white noise, we address the problem of synchronization among chaotic oscillators in the presence of common colored noise. We find that colored noise can induce synchronization, but the onset of synchronization, as characterized by the value of the critical noise amplitude above which synchronization occurs, can be different for noise of different colors. A formula relating the critical noise amplitudes among red, green, and white noise is uncovered, which holds for both complete and phase synchronization. The formula suggests practical strategies for controlling the degree of synchronization by noise, e.g., utilizing noise filters to suppress synchronization.

A tuneable attractor underlies yeast respiratory dynamics

Our understanding of the molecular structure and function in the budding yeast, Saccharomyces cerevisiae, surpasses that of all other eukaryotic cells. However, the fundamental properties of the complex processes and their control systems have been difficult to reconstruct from detailed dissection of their molecular components. Spontaneous oscillatory dynamics observed in self-synchronized continuous cultures is pervasive, involves much of the cellular network, and provides unique insights into integrative cell physiology. Here, in non-invasive experiments in vivo, we exploit these oscillatory dynamics to analyse the global timing of the cellular network to show the presence of a low-order chaotic component. Although robust to a wide range of environmental perturbations, the system responds and reacts to the imposition of harsh environmental conditions, in this case low pH, by dynamic re-organization of respiration, and this feeds upwards to affect cell division. These complex dynamics can be represented by a tuneable attractor that orchestrates cellular complexity and coherence to the environment.

Characterization of noise-induced strange nonchaotic attractors

Strange nonchaotic attractors (SNAs) were previously thought to arise exclusively in quasiperiodic dynamical systems. A recent study has revealed, however, that such attractors can be induced by noise in nonquasiperiodic discrete-time maps or in periodically driven flows. In particular, in a periodic window of such a system where a periodic attractor coexists with a chaotic saddle (nonattracting chaotic invariant set), none of the Lyapunov exponents of the asymptotic attractor is positive. Small random noise is incapable of causing characteristic changes in the Lyapunov spectrum, but it can make the attractor geometrically strange by dynamically connecting the original periodic attractor with the chaotic saddle. Here we present a detailed study of noise-induced SNAs and the characterization of their properties. Numerical calculations reveal that the fractal dimensions of noise-induced SNAs typically assume fractional values, in contrast to SNAs in quasiperiodically driven systems whose dimensions are integers. An interesting finding is that the fluctuations of the finite-time Lyapunov exponents away from their asymptotic values obey an exponential distribution, the generality of which we are able to establish by a theoretical analysis using random matrices. We suggest a possible experimental test. We expect noise-induced SNAs to be general.

Effect of noise on generalized chaotic synchronization

When two characteristically different chaotic oscillators are coupled, generalized synchronization can occur. Motivated by the phenomena that common noise can induce and enhance complete synchronization or phase synchronization in chaotic systems, we investigate the effect of noise on generalized chaotic synchronization. We develop a phase-space analysis, which suggests that the effect can be system dependent in that common noise can either induce/enhance or destroy generalized synchronization. A prototype model consisting of a Lorenz oscillator coupled with a dynamo system is used to illustrate these phenomena.

Stochastic bifurcation in a driven laser system: experiment and theory

We analyze the effects of stochastic perturbations in a physical example occurring as a higher-dimensional dynamical system. The physical model is that of a class- B laser, which is perturbed stochastically with finite noise. The effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior, or noise-induced chaos. To analyze the qualitative change, we apply the technique of the stochastic Frobenius-Perron operator [L. Billings et al., Phys. Rev. Lett. 88, 234101 (2002)] to a model of the experimental system. Our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment. In quantifying the stochastic bifurcation, we have computed a transition matrix describing the probability of transport from one region of phase space to another, which approximates the stochastic Frobenius-Perron operator. This mechanism depends on both the standard deviation of the noise and the global topology of the system. Our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise-induced chaos in both theory and experiment.

Multi-scale continuum mechanics: from global bifurcations to noise induced high-dimensional chaos

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.

Experimental characterization of transition to chaos in the presence of noise

Transition to chaos in the presence of noise is an important problem in nonlinear and statistical physics. Recently, a scaling law has been theoretically predicted which relates the Lyapunov exponent to the noise variation near the transition. Here we present experimental observation of noise-induced chaos in an electronic circuit and obtain the fundamental scaling law characterizing the transition. The experimentally obtained scaling exponent agrees very well with that predicted by theory.