Synchronization of self-sustained oscillators by common white noise

We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators.     

Response of a gyrotron to small-amplitude low-frequency-modulated microwaves reflected from a plasma

Experiments in the L-2M stellarator revealed the intense noise modulation of the gyrotron power and the change in its mean value under the action of the noise modulation of radiation reflected from the plasma column. The effect observed is explained in terms of the resonant locking of the gyrotron self-oscillations due to wave reflection from the fluctuating plasma load.     

Invariant power law distribution of langevin systems with colored multiplicative noise

The random multiplicative process is studied for the case of a colored multiplicative noise with exponentially decreasing autocorrelation function. We observe the power law exponent of probability distribution in a statistically steady state numerically to clarify the effect of finite correlation time. The renormalization procedure is applied to derive the power law exponent theoretically. The power law exponent is inversely proportional to the autocorrelation time of the multiplicative noise.     

Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

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

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

Lyapunov exponent of the random Schrödinger operator with short-range correlated noise potential

We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.     

Effect of noise on the neutral direction of chaotic attractor

A chaotic attractor from a deterministic flow must necessarily possess a neutral direction, as characterized by a null Lyapunov exponent. We show that for a wide class of chaotic attractors, particularly those having multiple scrolls in the phase space, the existence of the neutral direction can be extremely fragile in the sense that it is typically destroyed by noise of arbitrarily small amplitude. A universal scaling law quantifying the increase of the Lyapunov exponent with noise is obtained. A way to observe the scaling law in experiments is suggested.     

Stochastic resonance of a periodically driven neuron under non-Gaussian noise

We investigate the first-passage-time statistics of the integrate-fire neuron model driven by a sub-threshold harmonic signal superposed with a non-Gaussian noise. Here, we considered the noise as the result of a random multiplicative process displaced from the origin by an additive term. Such a mechanism generates a power-law distributed noise whose characteristic decay exponent can be finely tuned. We performed numerical simulations to analyze the influence of the noise non-Gaussian character on the stochastic resonance condition. We found that when the noise deviates from Gaussian statistics, the resonance condition occurs at weaker noise intensities, achieving a minimum at a finite value of the distribution function decay exponent. We discuss the possible relevance of this feature to the efficiency of the firing dynamics of biological neurons, as the present result indicates that neurons would require a lower noise level to detect a sub-threshold signal when its statistics departs from Gaussian.     

Chaos Control for Coupling of the Double-Well Duffing System Based on Random Phase Disturbance

Non-feedback methods of chaos control are suited for practical applications. For possible practical applications of the control methods, the robustness of the methods in the presence of noise is of special interest. The noise can be in the form of external disturbances to the system or in the form of uncertainties due to inexact model of the system. This paper deals with the effect of random phase disturbance for a class of coupling of the Double-Well Duffing system in the presence of the noise. Lyapunov index is an important indicator to describe chaos. When the sign of the top Lyapunov exponent is positive, the system is chaotic. We compute top Lyapunov exponent by the Khasminskii’s transform formula of spherical coordinate and extension of Wedig’s algorithm based on linear stochastic system. With the change of the average of top Lyapunov exponent sign, we show that random phase can suppress chaos. Finally Poincaré map and phase portraits analysis are studied to confirm the obtained results.     

Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors

We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.     

The largest Lyapunov exponent of chaotic dynamical system in scale space and its application

The largest Lyapunov exponent is an important invariant of detecting and characterizing chaos produced from a dynamical system. We have found analytically that the largest Lyapunov exponent of the small-scale wavelet transform modulus of a dynamical system is the same as the system's largest Lyapunov exponent, both discrete map and continuous chaotic attractor with one or two positive Lyapunov exponents. This property has been used to estimate the largest Lyapunov exponent of chaotic time series with several kinds of strong additive noise.     

A Stochastic Nonlinear Schrödinger Equation¶with Multiplicative Noise

We study a conservative stochastic nonlinear Schr?dinger equation with a multiplicative noise. We show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities, the critical exponent being the same, in dimension 1 or 2, as the critical exponent of the deterministic equation. Received: 21 December 1998 / Accepted: 22 February 1999     

Generation of self-oscillations from a singly resonant periodically poled potassium titanyl phosphate crystal frequency doubler

We report on the generation of self-oscillations from a continuously pumped singly resonant frequency doubler based on a periodically poled potassium titanyl phosphate crystal （PPKTP）. The sustained square-wave and staircase curve of self-oscillations are obtained when the incident pump powers are below and above the threshold of subharmonic-pumped parametric oscillation （SPO）, respectively. The self-oscillations can be explained by the competition between the phase shifts induced by cascading nonlinearity and thermal effect, and the influence of fundamental nonlinear phase shift by the generation of SPO. The simulation results are in good agreement with the experiment data.     

Renormalization group analysis of some dynamical systems with noise

We formulate a renormalization group analysis for the study of the accumulation of period doubling in the presence of noise. The main tool is a renormalization of the time evolution of the noise. The critical indices depend on the nature of the noise, but are given by thermodynamic quantities describing the large deviations of the Lyapunov exponent of the linearized random renormalization.     

Multifractal behaviour of n-simplex lattice

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, β _L, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex lattices.     

Effect of noise on chaos synchronization in time-delayed systems: Numerical and experimental observations

We discuss the constructive role of noise (white and colored) in chaos synchronization in time-delayed systems. We first numerically investigate noise-induced synchronization (NIS) between two identical uncoupled Ikeda and Mackey–Glass systems. We find that synchronization occurs above a critical noise intensity that differs for different colors of noise. Synchronization onset is characterized by the value of the maximum transverse Lyapunov exponent. We then discuss the enhancement of chaos synchronization between two time-delayed systems when they are coupled unidirectionally. The effect of parameter mismatch for NIS is described in detail. We provide experimental evidence of NIS for a Mackey–Glass-like system in an electronic circuit using different colors of noise. An integration scheme for time-delayed systems in the presence of additive white and colored noise is discussed.     

On the coherent self-oscillation mechanism under transverse breakdown in n-InSb (T = 77 K)

A nonlinear theory of self-oscillation in crossed electric and magnetic fields is worked out. Effects of the magnetic field intensity, sample dimensions, and surface recombination rate on the generation conditions of coherent (monochromatic) self-oscillations are studied. A domain model of coherent self-oscillations excited under transvere breakdown and strong magnetoconcentration effect is proposed. Numerical calculations for n-InSb (T = 77 K) are carried out. Coherent self-oscillations are shown to be induced for frequencies ～ 10¹⁰ Hz.     

Noise induced destruction of zero Lyapunov exponent in coupled chaotic systems

Recently, it has been found that noise can induce chaos and destruct the zero Lyapunov exponent in the situation where a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window [Phys. Rev. Lett. 88 (2002) 124101]. Here we report that noise can also destruct the zero Lyapunov exponent in coupled chaotic systems where there is only one attractor. Moreover, the zero Lyapunov exponent in noise free will become positive when adding noise and be proportional to the average frequency of bursting induced by noise. A physical theory and numerical simulations are presented to explain how the average frequency of bursting depends on the coupling and noise strength.     

Stability of a nonlinear oscillator with random damping

A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation. In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.     

Effect of thermal fluctuations on the spin self-oscillations in a microwave nanosized oscillator

Reduced equations are derived for the slow evolution of the phase and amplitude of spin self-oscillations arising in a free magnetic layer of a nanosized microwave oscillator under the action of an over-threshold spin-polarized current. These equations are used to calculate the spectral intensity of the fundamental-harmonic signal of a spin self-oscillator allowing for thermal noise in the spin subsystem. It is shown that the line width as a function of the current is determined by a bifurcation-type variation in the spin state near the self-oscillation threshold. The self-oscillation nonisochronism makes the spectral line profile asymmetric. The line-profile asymmetry can change as the cycle-appearance point is approached.