Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

Stochastic Perturbations to Dynamical Systems: A Response Theory Approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.     

Synchronization of noisy systems by stochastic signals

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.     

Stochastic multi-resonance in an overdamped bistable system with two types of modulation signal

The stochastic resonance (SR) phenomenon in an overdamped bistable system with multiplicative and additive noise is investigated. The signal-to-noise ratio (SNR) is calculated when two types of modulation signal are added to the system. The effects of the intensities, the frequencies and relative phase shift of two types of modulation signal on the SNR are discussed, respectively. Research results show that: (i) the intensities of two types of modulation signal can enhance the maximum in the SNR as a function of the noise intensity, and the frequencies can restrain it; (ii) the additive modulation signal can enhance the maximum in the SNR as a function of the noise intensity in comparison with the multiplicative modulation signal; (iii) when both modulation frequencies are equal, the SNR as a function of the relative phase shift exhibits multiple maxima. The multiple maxima in the SNR identifies the characteristic of the stochastic multi-resonance phenomenon.     

Self-generated dynamic landscape: The message–receiver interaction case

In the present work we present a model, based on a particular differential stochastic equation, to study the interaction between an incoming message and its interpreter. The particular stochastic dynamic used to understand such process is written using a delayed Langevin equation with white noise. The results of this kind of interaction can be understood in a general framework that we name the self generated dynamic landscape.     

Anisotropic stochastic resonance in the system with a 1/f spectrum

A new behavior of two-dimensional system equations has been discovered upon stochastic resonance: at low frequencies of the harmonic excitation and increasing intensity of the white noise, the motion of the system is limited to two mutually perpendicular directions. In this case we obtain an anisotropic stochastic resonance. With further increase in the intensity of the white noise in the system, one can observe a normal stochastic resonance in which an increase in the intensity of fluctuations in the frequency range measured is blurred by directions.     

二值噪声激励下欠阻尼周期势系统的随机共振

研究了二值噪声和周期信号共同激励下欠阻尼周期势系统的随机共振. 利用随机能量法计算了系统的平均输入能量和平均输出信号的振幅和相位差, 讨论了二值噪声对随机共振的影响. 发现随着噪声强度的增大, 平均输入能量曲线存在一个极小值和一个极大值, 系统出现先抑制后共振的现象; 同时, 系统信噪比曲线随噪声强度的增加出现单峰现象, 说明系统存在随机共振现象.     

A theory of open systems based on stochastic differential equations

For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.     

四稳系统的双重随机共振特性

提出了一类8次势函数并讨论了其分岔特性,得到由左、右2个小尺度双稳势和中间势垒构成的对称四稳系统.建立了在周期力和随机力共同作用下四稳系统输出响应的近似解析表达式,并从能量角度引入功这一过程量来刻画大、小不同尺度双稳势之间的作功能力,发现四稳势中存在着双重随机共振现象.理论分析与数值仿真结果表明,当中间势垒高度大于左右...     

Fokker-Planck Equation of a Stochastic System Driven by Spatially-Related Noises

In this paper we study a general stochastic system driven by the spatially-related Gaussian white noises. The corresponding Fokker-Planck equation is calculated; and some typical cases are analyzed. Finally, by the Fokker-Planck equation derived in the paper we study a single bistable kinetic process with spatially-related noise. The results obtained in the paper provide a correct foundation for the.treatment of the stochastic systems driven by spatially-related noises.     

Adaptive Stochastic Resonance-Based Processing of Weak Magnetic Slippage Signals of Bearings

Slip is one of the most common forms of failure in aviation bearings, and it can pose a great threat to the stable operation of aviation bearings. Bearing cage speed monitoring methods based on weak magnetic detection can achieve nondestructive measurements. However, the method suffers from solid signal background noise due to the high sensitivity of the sensor. Therefore, in this paper, an adaptive stochastic resonance algorithm was proposed in response to the characteristics of the weak magnetic detection signal and the problem of solid noise. In addition, by adaptively adjusting the coefficients of the stochastic resonance system—by an improved moth flame optimization algorithm—the drawback in which the stochastic resonance method required artificially set parameters for extracting the feature frequencies of the weak magnetic signals was solved. In this process, we used parameters, such as general refined composite multi-scale sample entropy, as the adaptation function of the optimization algorithm. In the end, simulation and experimental outcomes verified the efficacy of the approach put forward.     

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.     

Stochastic Resonance in Nonequilibrium Single-Stable System

The harmonic noise is used as an external stochastic source in non-Markovian equilibrium system. A bi-peak resonance phenomenon is carried out due to the external nohe frequency close to the frequencies of damped linear-oscillatpr and internal noise, respectively.     

Influence of inerter on natural frequencies of vibration systems

This paper investigates the influence of inerter on the natural frequencies of vibration systems. First of all, the natural frequencies of a single-degree-of-freedom (SDOF) system and a two-degree-of-freedom (TDOF) system are derived algebraically and the fact that the inerter can reduce the natural frequencies of these systems is demonstrated. Then, to further investigate the influence of inerter in a general vibration system, a multi-degree-of-freedom system (MDOF) is considered. Sensitivity analysis is performed on the natural frequencies and mode shapes to demonstrate that the natural frequencies of the MDOF system can always be reduced by increasing the inertance of any inerter. The condition for a general MDOF system of which the natural frequencies can be reduced by an inerter is also derived. Finally, the influence of the inerter position on the natural frequencies is investigated and the efficiency of inerter in reducing the largest natural frequencies is verified by simulating a six-degree-of-freedom system, where a reduction of more than 47 percent is obtained by employing only five inerters.     

Experimental analysis of mode-hopping in bulk semiconductor lasers

In this work we experimentally study mode-hopping in bulk semiconductor lasers. This stochastic process is ruled by Kramers statistics with a decay rate depending on the laser parameters of the temperature of the substrate and the pumping current. For a general combination of these parameters the average residence times in the two active modes are not equal, resulting in an asymmetric probability distribution for the modal intensities. We show that, by choosing an appropriate path in the parameter space, we can vary the residence times of the two modes while holding their ratio constant. Along this path, the shape of modal intensities distributions are constant up to a scaling factor which is a function of the laser parameters. Then, the system can be described by a single Langevin equation. The effect of adding noise to the pumping current is also investigated. PACS 42.65.Sf; 42.55.Sa; 42.50.-p     

双稳随机动力系统信号调制噪声效应的数值分析

用数值方法研究了双稳随机动力系统的信号调制噪声效应.结果表明，正弦信号在系统输出中的效应仍为正弦信号，白噪声的效应则为维纳过程，通过选择合适的系统参数，可以减小系统输出中信号和噪声之间的耦合效应，系统可以大大抑制噪声，从而在双稳系统中可以产生信号调制噪声效应. 关键词： 双稳系统 信号调制噪声效应 随机共振     

周期外力对频率涨落的过阻尼谐振子所作的功和能量随机共振

研究了周期外力对频率涨落的过阻尼谐振子系统作功的特点. 结果揭示了瞬时功率随时间的周期变化出现不对称性. 研究结果还揭示周期外力一个周期对系统所做的功随乘法噪声强度的变化出现非单调行为, 系统是否出现能量随机共振与抑制并存, 由乘法噪声与加法噪声之间互关联的符号决定. 关键词： 过阻尼谐振子 频率涨落 周期外力作功 能量随机共振与抑制     

The Stochastic stability of a Logistic model with Poisson white noise

The stochastic stability of a logistic model subjected to the effect of a random natural environment, modeled as Poisson white noise process, is investigated. The properties of the stochastic response are discussed for calculating the Lyapunov exponent, which had proven to be the most useful diagnostic tool for the stability of dynamical systems. The generalised It? differentiation formula is used to analyse the stochastic stability of the response. The results indicate that the stability of the response is related to the intensity and amplitude distribution of the environment noise and the growth rate of the species.     

Stochastic resonance in a time-delayed asymmetric bistable system with mixed periodic signal

This paper studies the phenomenon of stochastic resonance in an asymmetric bistable system with time-delayed feedback and mixed periodic signal by using the theory of signal-to-noise ratio in the adiabatic limit. A general approximate Fokker--Planck equation and the expression of the signal-to-noise ratio are derived through the small time delay approximation at both fundamental harmonics and mixed harmonics. The effects of the additive noise intensity $Q$, multiplicative noise intensity $D$, static asymmetry $r$ and delay time $\tau$ on the signal-to-noise ratio are discussed. It is found that the higher mixed harmonics and the static asymmetry $r$ can restrain stochastic resonance, and the delay time $\tau $ can enhance stochastic resonance. Moreover, the longer the delay time $\tau $ is, the larger the additive noise intensity $Q$ and the multiplicative noise intensity $D$ are, when the stochastic resonance appears.