Deformation of limit cycle under perturbations

The robustness of limit cycles of nonlinear dynamical systems is investigated by adding a small random velocity field to the famous van der Pol (VDP) equation in its two-dimensional phase plane. Our numerical calculations show that a limit cycle does not change much under a weak random perturbation. Thus it confirms the conjecture that a limit cycle will make only a small deformation under an external perturbation. The idea can be used to understand the ac response of self-sustained oscillations in nonlinear dynamical systems.     

Chaotic behaviour and limit cycle behaviour of anharmonic systems with periodic external perturbations

The chaotic behaviour and limit cycle behviaiour of the dynamical system $\text{x}\text{?}\text{+ α}\text{x}\text{?}\text{+}\text{d}\text{V(x)}\text{d}\text{x}\text{= ?}\text{cos}\text{(Ωt)}$ is investigated for various potentials V and parameters values α, $\text{?}$ and $\text{Ω}$.     

Effect of noise on coupled chaotic systems

The effect of noise in inducing order on various chaotically evolving systems is reviewed, with special emphasis on systems consisting of coupled chaotic elements. In many situations it is observed that the uncoupled elements when driven by identical noise, show synchronization phenomena where chaotic trajectories exponentially converge towards a single noisy trajectory, independent of the initial conditions. In a random neural network, with infinite range coupling, chaos is suppressed due to noise and the system evolves towards a fixed point. Spatiotemporal stochastic resonance phenomenon has been observed in a square array of coupled threshold devices where a temporal characteristic of the system resonates at a given noise strength. In a chaotically evolving coupled map lattice with the logistic map as local dynamics and driven by identical noise at each site, we report that the number ofstructures (a structure is a group of neighbouring lattice sites for values of the variable follow which the certain predefined pattern) follows a power-law decay with the length of the structure. An interesting phenomenon, which we callstochastic coherence, is also reported in which the abundance and lifetimes of these structures show characteristic peaks at some intermediate noise strength.     

Harmonic noise: Effect on bistable systems

The coordinate of a white noise driven harmonic oscillator is used as a stochastic source term in bistable dynamics. This new kind of Gaussian colored noise gives rise to resonance phenomena due to a peak in the spectrum. We investigate its effect on linear and bistable systems. We derive a Markovian approximation for driven bistable oscillators and overdamped systems. In the resonance region computer simulations were carried out using an extension of Fox' algorithm procedure for colored noise. We find an increase of the transition rates in bistable systems as compared with the case of bistable systems driven by white and exponentially correlated noise.     

Identification of fractional-order systems via a switching differential evolution subject to noise perturbations

In this Letter, a differential evolution variant, called switching DE (SDE), has been employed to estimate the orders and parameters in incommensurate fractional-order chaotic systems. The proposed algorithm includes a switching population utilization strategy, where the population size is adjusted dynamically based on the solution-searching status. Thus, this adaptive control method realizes the identification of fractional-order Lorenz, Lü and Chen systems in both deterministic and stochastic environments, respectively. Numerical simulations are provided, where comparisons are made with five other State-of-the-Art evolutionary algorithms (EAs) to verify the effectiveness of the proposed method.     

Models of chemical reaction systems with exactly evaluable limit cycle oscillations

Models of open chemical reaction systems with two variablesx, y and with elliptic limit cycles in concentration space are investigated. Thereby isothermal and homogeneous conditions and the validity of mass-action kinetics are assumed. Though we restrict ourselves to quadratic rate equations a lot of systems with this behaviour is found; all based on the trimolecular reactionA + 2X 3X. Both globally stable and only locally stable limit cycles are possible. The time dependence of the variables on the cycle is exactly evaluated for all these systems. Bifurcations of the limit cycle are described analytically.D 82 (Diss. TH Aachen)     

Effect of noise on characteristics of optical waveguide systems

Transient regimes of an optical waveguide system in the presence of additive and multiplicative noises are studied. The presence of noise is shown to result in a transformation of the spectrum of radiation modes. The effect of noise on the efficiency of input of electromagnetic energy into a waveguide system is studied.     

非马尔科夫耗散系统长时演化下的极限环振荡现象

本文研究结构化环境中非马尔科夫耗散系统在长时演化下可能出现的极限环振荡现象. 对于欧姆型谱密度环境中的二能级系统, 由于体系只允许一个束缚态模, 给定初态系统在Bloch空间的长时演化将收敛于一个极限环. 研究揭示了极限环半径与环心位置同环境谱密度函数间的关系. 对于多带光子晶体环境中的二能级系统, 由于其可以存在多个束缚态, 研究展现了系统在长时演化下可能出现的收敛于环面或周期或准周期的振荡行为. 有关环面的特征量与环境谱密度间的量化关系同样得以刻画. 论文随后讨论了两比特系统关联量在局域非马尔科夫耗散环境中长时演化可能出现的特征行为.     

Asymptotic properties of coupled nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle

We apply the singular perturbation technique, developed in the companion paper, to the study of the fluctuations at the onset of a limit cycle, both for the cases of a soft and a hard transition. The technique and results are illustrated on the Poincaré model (soft transition) and on the Van der Pol oscillator (hard transition).     

Quantumq-white noise and aq-central limit theorem

We establish a connection between the Azéma martingales and certain quantum stochastic processes with increments satisfyingq-commutation relations. This leads to a theory ofq-white noise onq-*-bialgebras and to a generalization of the Fock space representation theorem for white noise on *-bialgebras. In particular, quantum Azéma noise,q-interpolations between Fermion and Boson quantum Brownian motion and unitary evolutions withq-independent multiplicative increments are studied. It follows from our results that the Azéma martingales and theq-interpolations are central limits of sums ofq-independent, identically distributed quantum random variables.     

Effect of phase noise on optical minimum-shift keying transmission systems

Phase-modulated (PM) signals have recently been used in long-haul lightwave communication systems to reach record distances. Added directly to the signal phase, linear and nonlinear phase noise cause performance degradation in PM systems. Minimum-shift keying (MSK), as a special format of PM signals, presents some different features of phase noise tolerance. This paper investigates the effect of phase noise on the performance degradation in the 10 Gb/s optical MSK systems and analyzes the contribution of phase noise to the bit error rate (BER), compared with the conventional PM format, 50% duty cycle optical return-to-zero differential phase-shift keying (RZ-DPSK) signals, in several typical local fiber dispersion transmission systems. The effect of intensity noise on the performance degradation in the corresponding transmission systems is also investigated.     

Modulation bandwidth and noise limit of photoconductive gates

The modulation bandwidth and noise limit of a photoconductive sampling gate are studied by reducing the parasitic capacitance and leakage current of the sampling circuit using an integrated junction field-effect transistor (JFET) source follower. The modulation bandwidth of the photoconductive sampling gate is limited by the external parasitic capacitance, and its efficiency is found to saturate at a laser gating power of about 1 mW. It is determined that the noise of the photoconductive sampling gate is dominated by the photovoltaic current due to the gating laser amplitude fluctuation. A minimum noise level of 4 nV Hz^–1/2 has been measured, and an enhancement in signal-to-noise ratio by a factor of >45 has been achieved after the integration of the source follower with the photoconductive sampling gate. The JFET source follower serves to increase the modulation bandwidth of the photoconductive sampling gate by about 15 times and buffer the charge of the measured signal using its extremely high gate input impedance. The performance of the photoconductive sampling gate in regard to invasiveness and gating efficiency has been optimized, while a picosecond temporal resolution has been maintained and the signal-to-noise performance has been enhanced using a gating laser power as low as 10 W.     

Effect of dispersion on nonlinear phase noise in optical transmission systems

An analytic expression for the variance of nonlinear phase noise that uses a first-order perturbation technique is obtained. The results show that for highly dispersive transmission systems, amplified spontaneous emission-induced phase noise due to self-phase modulation becomes much smaller than that for the systems with no dispersion.     

Periodic solutions and perturbations of dynamical systems

We deal with some problems concerning periodic solutions of perturbed dynamical systems. Sufficient conditions for the existence of periodic solution(s) of perturbed system are obtained. Moreover, we derive some properties of the set of all perturbed terms of a dynamical system under which the perturbed system has periodic solution(s). The method is based on the analysis of the space of all solutions of a nonperturbed dynamical system.     

Strong perturbations in nonlinear systems

A novel case of probabilistic coupling for hybrid stochastic systems with chaotic components via Markovian switching is presented. We study its stability in the norm, in the sense of Lyapunov and present a quantitative scheme for detection of stochastic stability in the mean. In particular we examine the stability of chaotic dynamical systems in which a representative parameter undergoes a Markovian switching between two values corresponding to two qualitatively different attractors. To this end we employ, as case studies, the behaviour of two representative chaotic systems (the classic Rössler and the Thomas-Rössler models) under the influence of a probabilistic switch which modifies stochastically their parameters. A quantitative measure, based on a Lyapunov function, is proposed which detects regular or irregular motion and regimes of stability. In connection to biologically inspired models (Thomas-Rössler models), where strong fluctuations represent qualitative structural changes, we observe the appearance of stochastic resonance-like phenomena i.e. transitions that lead to orderly behavior when the noise increases. These are attributed to the nonlinear response of the system.     

Chaotic response of a limit cycle

External periodic modulation of a nonlinear oscillator may lead to chaotic behavior. This phenomenon is attributed to the existence of a strange attractor, which embodies essentially a folding motion as is met within the Bernoulli shift or the baker's transformation. The results obtained for the Brussels model are discussed from this viewpoint.     

Sensitivity analysis of limit cycle oscillations

Many unsteady problems equilibrate to periodic behavior. For these problems the sensitivity of periodic outputs to system parameters are often desired, and must be estimated from a finite time span or frequency domain calculation. Sensitivities computed in the time domain over a finite time span can take excessive time to converge, or fail altogether to converge to the periodic value. Additionally, finite span outputs can exhibit local extrema in parameter space which the periodic outputs they approximate do not, hindering their use in optimization. We derive a theoretical basis for this error and demonstrate it using two examples, a van der Pol oscillator and vortex shedding from a low Reynolds number airfoil. We show that output windowing enables the accurate computation of periodic output sensitivities and may allow for decreased simulation time to compute both time-averaged outputs and sensitivities. We classify two distinct window types: long-time, over a large, not necessarily integer number of periods; and short-time, over a small, integer number of periods. Finally, from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation, respectively.     

Effect of detector noise in incoherent hybrid imaging systems

Hybrid imaging systems involve the joint design of an optical image-gathering module and digital processing algorithms to obtain a required final image. They have the potential to achieve imaging performance hitherto unobtainable by conventional imaging techniques. A reduction in the signal-to-noise ratio of the final image is one of their main disadvantages when one is considering linear signal processing. We analyze the effect of additive white noise at the detector on the performance of hybrid imaging systems under quasi-monochromatic incoherent illumination. We also show numerical results and computer-simulated images for an extended depth-of-field hybrid system.