Characterization of Stochastic Bifurcations in a Simple Biological Oscillator

This study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first-passage-time problem of the Ornstein–Uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density. Stochastic phase-locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase. A firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel.     

Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach

Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. The probability density functions of the first-passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator.     

A note on stochastic phase lockings in a relaxation oscillator forced by a sine input with additive noise

Summary Noise effects on phase lockings in a system consisting of a piecewise-linear van der Pol relaxation oscillator driven by a periodic input are studied. The problem of finding the period of the oscillator is reduced to the first-passage-time problem of the Ornstein-Uhlenbeck process with time-varying boundary. Using the probability density functions of the first-passage time, the operator which governs a transition of an input phase density after one cycle of the oscillator is defined. Phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

Stochastic period-doubling bifurcation in biharmonic driven Dulling system with random parameter

Stochastic period-doubling bifurcation is explored in a forced Duffing system with a bounded random parameter as an additional weak harmonic perturbation added to the system. Firstly, the biharmonic driven Duffing system with a random parameter is reduced to its equivalent deterministic one, and then the responses of the stochastic system can be obtained by available effective numerical methods. Finally, numerical simulations show that the phase of the additional weak harmonic perturbation has great influence on the stochastic period-doubling bifurcation in the biharmonic driven Duffing system. It is emphasized that, different from the deterministic biharmonic driven Duffing system, the intensity of random parameter in the Duffing system can also be taken as a bifurcation parameter, which can lead to the stochastic period-doubling bifurcations.     

Stochastic dynamics of parametrically excited two d.o.f. systems with symmetry

In this paper we investigate the non-linear dynamics of a two-degree-of-freedom system with symmetries subject to random parametric excitation. The study of this non-linear near-Hamiltonian system is simplified by using the symmetry and separation of scales present in the problem. To this end, we study the equations as a random perturbation of a four-dimensional weakly dissipative Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced process is simply a Markov process on a line. Examination of the reduced Markov process on the line yields many important results, namely, probability density functions, and stochastic bifurcations. The steady state dynamics is computed explicitly. Phenomenological and dynamical bifurcations are investigated. The approach adopted in this paper can in principle, be applied to any four-dimensional integrable system.     

Luenberger-Type Observer Design for Stochastic Time-Delay Systems

This paper deals with the problem of an observer design for stochastic time-delay systems. The system states are unmeasured. We derive delay-dependent LMI criteria by means of the Leibniz-Newton formula, the Itô’s differential operator and stochastic Lyapunov stability theory in order to obtain sufficient conditions for the asymptotic stability in the mean square for the closed-loop stochastic time-delay system. The proposed conditions are easily and numerically tractable via a Matlab LMI toolbox. The effectiveness of the control strategy is verified by numerical experiments.     

Quantum Stochastic Convolution Cocycles II

Schürmann’s theory of quantum Lévy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Lévy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Lévy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.     

Stochastic quantization and detailed balance in Fokker-Planck dynamics

The path integral and operator formulations of the Fokker-Planck equation are considered as stochastic quantizations of underlying Euler-Lagrange equations. The operator formalism is derived from the path integral formalism. It is proved that the Euler-Lagrange equations are invariant under time reversal if detailed balance holds and it is shown that the irreversible behavior is introduced through the stochastic quantization. To obtain these results for the nonconstant diffusion Fokker-Planck equation, a transformation is introduced to reduce it to a constant diffusion Fokker-Planck equation. Critical comments are made on the stochastic formulation of quantum mechanics.     

Stochastic Aspects of Cardiac Arrhythmias

Abnormal cardiac rhythms (cardiac arrhythmias) often display complex changes over time that can have a random or haphazard appearance. Mathematically, these changes can on occasion be identified with bifurcations in difference or differential equation models of the arrhythmias. One source for the variability of these rhythms is the fluctuating environment. However, in the neighborhood of bifurcation points, the fluctuations induced by the stochastic opening and closing of individual ion channels in the cell membrane, which results in membrane noise, may lead to randomness in the observed dynamics. To illustrate this, we consider the effects of stochastic properties of ion channels on the resetting of pacemaker oscillations and on the generation of early afterdepolarizations. The comparison of the statistical properties of long records showing arrhythmias with the predictions from theoretical models should help in the identification of different mechanisms underlying cardiac arrhythmias.     

Influence of Stochastic High-Order Perturbations on Soliton Transmission System

The influence of stochastic high-order perturbations on soliton transmission system is studied by self-companying operator method. Arrival time jitters variation induced by stochastic third-order dispersion is approximately proportional to three powers of transmission distance, that by stochastic nonlinear dispersion and self-steeping are approximately proportional to the transmission distance.     

Stochastic differential equation derivation: Comparison of the Markov method versus the additive method

There are several methods of transforming an ordinary differential equation into a stochastic differential equation (SDE). The two most common are adding noise to a system parameter or variable and transforming to a SDE or deriving the SDE by assuming an underlying Markov process. Using simple one- and two-dimensional systems we investigate the differences in dynamics and bifurcations between SDE derived by each method from simple deterministic population models.     

Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation

A response analysis procedure is developed for a vibro-impact system excited by colored noise. The non-smooth transformation is used to convert the vibro-impact system into a new system without impact term. With the help of the modified quasi-conservative averaging, the total energy of the new system can be approximated as a Markov process, and the stationary probability density function （PDF） of the total energy is derived. The response PDFs of the original system are obtained using the analytical solution of the stationary PDF of the total energy. The validity of the theoretical results is tested through comparison with the corresponding simulation results. Moreover, stochastic bifurcations are also explored.     

Synchronization of multi-frequency noise-induced oscillations

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.     

Perturbation Theory for Continuous Stochastic Equations

The various general perturbational schemes for continuous stochastic equations are considered. These schemes have many analogous features with the iterational solution of Schwinger equation for S-matrix. The following problems are discussed: continuous stochastic evolution equations for probaibility distribution functionals, evolution equations for equal time correlators, perturbation theory for Gaussian and Poissonian additive noise, perturbation theory for birth and death processes. stochastic properties of systems with multiplicative noise. The general results are illustrated by diffusion - controlled reactions, fluctuations in closed systems with chemical processes, propagation of waves in random media in parabolic equation approximation, and nonequilibrium phase transitions in systems with Poissonian breeding centers. The rate of irreversible reaction X + X → A (Smoluchowski process) is calculated with the use of general theory based on continuous stochastic equations for birth and death processes. The threshold criterion and range of fluctuational region for synergetic phase transition in system with Poissonian breeding centers are also considered.     

一般阻尼随机微分方程的拟局部振荡算法

提出了一种模拟随机微分方程的拟局部振荡算法,即利用算符劈裂方法和势函数的泰勒展开,对噪声作用下耗散粒子的时间演化算符进行分解,得到了对应涨落行为的扩散算符和对应确定轨迹的漂移算符.其中局部简谐势场的涨落过程可获得解析解,而剩余的确定项则利用简单的Euler算法积分.应用到几个算例并与常用的两种算法相比较,结果表明:本算法随时间步长最稳定,可使用较大的时间步长.     

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

We consider the Hamiltonian operator associated to the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe a quantum mechanical evolution in the presence of a “quantum noise”. We characterize such a Hamiltonian in the case of arbitrary multiplicity and bounded coefficients: we find an essentially self-adjoint restriction of the operator and, in particular, we provide an explicit construction of a dense set of vectors belonging to its domain. An erratum to this article is available at .     

A history-dependent stochastic predator-prey model: Chaos and its elimination

A non-Markovian stochastic predator-prey model is introduced in which the prey are immobile plants and predators are diffusing herbivors. The model is studied by both mean-field approximation (MFA) and computer simulations. The MFA results a series of bifurcations in the phase space of mean predator and prey densities, leading to a chaotic phase. Because of emerging correlations between the two species distributions, the interaction rate alters and if it is chosen to be the value which is obtained from the simulation, then the chaotic phase disappears. Received 12 July 1999