有界噪声和谐和激励联合作用下一类非线性系统的混沌研究

研究一类有界噪声和谐和激励联合作用下的非线性系统，首先用多尺度方法将该系统约化，针对约化后的平均系统，利用随机Melnikov过程方法结合均方值准则导出随机系统可能产生混沌运动的临界条件，结果表明在一定的参数范围内，随着Weiner过程强度参数值的增大，混沌的临界激励幅值先递减继而递增. 同时，用两类数值方法即最大Lyapunov指数法和Poincare截面法验证了解析结果. 关键词： 有界噪声 多尺度 随机Melnikov过程 混沌     

Duffing系统随机相位抑制混沌与随机共振并存现象的机理研究

针对随机相位作用的Duffing混沌系统, 研究了随机相位强度变化时系统混沌动力学的演化行为及伴随的随机共振现象. 结合Lyapunov指数、庞加莱截面、相图、时间历程图、功率谱等工具, 发现当噪声强度增大时, 系统存在从混沌状态转化为有序状态的过程, 即存在噪声抑制混沌的现象, 且在这一过程中, 系统亦存在随机共振现象, 而且随机共振曲线上最优的噪声强度恰为噪声抑制混沌的参数临界点. 通过含随机相位周期力的平均效应分析并结合系统的分岔图, 探讨了噪声对混沌运动演化的作用机理, 解释了在此过程中随机共振的形成机理, 论证了噪声抑制混沌与随机共振的相互关系.     

Noise-induced chaos in the electrostatically actuated MEMS resonators

In this Letter, nonlinear dynamic and chaotic behaviors of electrostatically actuated MEMS resonators subjected to random disturbance are investigated analytically and numerically. A reduced-order model, which includes nonlinear geometric and electrostatic effects as well as random disturbance, for the resonator is developed. The necessary conditions for the rising of chaos in the stochastic system are obtained using random Melnikov approach. The results indicate that very rich random quasi-periodic and chaotic motions occur in the resonator system. The threshold of bounded noise amplitude for the onset of chaos in the resonator system increases as the noise intensity increases.     

Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.     

Chaotic motion of Van der Pol–Mathieu–Duffing system under bounded noise parametric excitation

The chaotic behavior of Van der Pol–Mathieu–Duffing oscillator under bounded noise is investigated. By using random Melnikov technique, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in this system increases as the intensity of the noise in frequency increases, which is further verified by the maximal Lyapunov exponents of the system. The effect of bounded noise on Poincaré map is also investigated, in addition the numerical simulation of the maximal Lyapunov exponents.     

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.     

Noise can prevent onset of chaos in spatiotemporal population dynamics

Many theoretical approaches predict the dynamics of interacting populations to be chaotic but that has very rarely been observed in ecological data. It has therefore risen a question about factors that can prevent the onset of chaos by, for instance, making the population fluctuations synchronized over the whole habitat. One such factor is stochasticity. The so-called Moran effect predicts that a spatially correlated noise can synchronize the local population dynamics in a spatially discrete system, thus preventing the onset of spatiotemporal chaos. On the whole, however, the issue of noise has remained controversial and insufficiently understood. In particular, a well-built nonspatial theory infers that noise enhances chaos by making the system more sensitive to the initial conditions. In this paper, we address the problem of the interplay between deterministic dynamics and noise by considering a spatially explicit predator-prey system where some parameters are affected by noise. Our findings are rather counter-intuitive. We show that a small noise (i.e. preserving the deterministic skeleton) can indeed synchronize the population oscillations throughout space and hence keep the dynamics regular, but the dependence of the chaos prevention probability on the noise intensity is of resonance type. Once chaos has developed, it appears to be stable with respect to a small noise but it can be suppressed by a large noise. Finally, we show that our results are in a good qualitative agreement with some available field data.     

Geometry and quantum noise

We study the fine structure of long‐time quantum noise in correlation functions of AdS/CFT systems. Under standard assumptions of quantum chaos for the dynamics and the observables, we estimate the size of exponentially small oscillations and trace them back to geometrical features of the bulk system. The noise level is highly suppressed by the amount of dynamical chaos and the amount of quantum impurity in the states. This implies that, despite their missing on the details of Poincaré recurrences, ‘virtual’ thermal AdS phases do control the overall noise amplitude even at high temperatures where the thermal ensemble is dominated by large AdS black holes. We also study EPR correlations and find that, in contrast to the behavior of large correlation peaks, their noise level is the same in TFD states and in more general highly entangled states.     

非周期力在混沌控制中的双重功效

探讨了非周期力（有界噪声或混沌驱动力）在非线性动力系统混沌控制中的影响.以一类典型的含有五次非线性项的Duffing-van der Pol系统为范例,通过对系统的轨道、最大Lyapunov指数、功率谱幅值及Poincar截面的分析,发现适当幅值的有界噪声或混沌信号,一方面可以消除系统对初始条件的敏感依赖性,抑制系统的混沌行为,将系统的混沌吸引子转化为奇怪非混沌吸引子;另一方面也可以诱导系统的混沌行为,将系统的周期吸引子转化为混沌吸引子.从而揭示了非周期力在混沌控制中的双重功效：抑制混沌和诱导混沌. 关键词： 混沌控制 有界噪声 混沌驱动力     

多频谐和与噪声作用下Flickering振子的安全盆侵蚀与混沌

研究了催化反应Flickering振子在多频率确定性谐和外力和有界随机噪声联合作用下,系统安全盆的侵蚀和混沌现象.将Melnikov方法推广到包含有限多个频率外力和随机噪声联合作用的情形,推导出了系统的随机Melnikov过程,根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值,然后用数值模拟方法计算了系统的安全盆分岔点.结果表明,由于随机扰动的影响,系统的安全盆分岔点发生了偏移,并且使得混沌容易发生.同时证明,激励频率数目的增加扩大了参数空间上的混沌区域,也使得安全盆分岔提 关键词： 多频率激励 Flickering振子 安全盆 混沌     

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.     

Spatial period-doubling in open flow

Coupled logistic lattices with asymmetric coupling in space, with a fixed boundary condition at the left end, are investigated. The system shows a period-doubling bifurcation to chaos as a lattice point goes downflow. In contrast with usual period-doubling in low-dimensional systems, (i) no scaling behavior has been found, (ii) low noise is important for the bifurcation structures. The system corresponds to a model for an open flow, which may be of use for the study of the onset of turbulence in pipe flows.     

Feedback Induced Instability and Chaos in Semiconductor Lasers and Their Applications

Semiconductor laser with feedback is an excellent model for nonlinear optical system which shows chaotic dynamics. It is interesting not only from the fundamental physical study but also application standpoints of view. The dynamics of feedback induced instability and chaos, especially for optical feedback, and their applications are reviewed in this paper. The model of such a system is described by the laser rate equations. At first the dynamic behaviors of feedback induced instability and chaos in semiconductor lasers are discussed on the basis of the theory and experiments. Instability and chaos may be stabilized by the method of chaos control. Then we apply the method to suppress the noise induced by the feedback in a semiconductor laser. The synchronization of chaos between two similar systems is also an important issue in chaos applications and we discuss secure communications based on chaos synchronization. Some other examples of applications of feedback induced chaos are also described.     

Effects of randomness on chaos and order of coupled logistic maps

Natural systems are essentially nonlinear being neither completely ordered nor completely random. These nonlinearities are responsible for a great variety of possibilities that includes chaos. On this basis, the effect of randomness on chaos and order of nonlinear dynamical systems is an important feature to be understood. This Letter considers randomness as fluctuations and uncertainties due to noise and investigates its influence in the nonlinear dynamical behavior of coupled logistic maps. The noise effect is included by adding random variations either to parameters or to state variables. Besides, the coupling uncertainty is investigated by assuming tinny values for the connection parameters, representing the idea that all Nature is, in some sense, weakly connected. Results from numerical simulations show situations where noise alters the system nonlinear dynamics.     

相空间压缩实现时空混沌系统的广义同步

提出了一种通过相空间压缩实现时空混沌系统广义同步的方法. 以Fitzhugh-Nagumo反应扩散时空混沌系统为例,仿真模拟说明了该方法的有效性与实用性. 通过研究有界噪声作用下该系统的同步效果,表明这种同步方法具有较强的抗干扰能力. 此方法可以实现任意时空混沌系统的广义同步,具有普适性. 同步控制器结构简单、易于应用. 关键词： 时空混沌 广义同步 相空间压缩     

Infection dynamics in structured populations with disease awareness based on neighborhood contact history

It is well-known that the climate system, due to its nonlinearity, can be sensitive to stochastic forcing. New types of dynamical regimes caused by the noise-induced transitions are revealed on the basis of the classical climate model previously developed by Saltzman with co-authors and Nicolis. A complete parametric classification of dynamical regimes of this deterministic model is carried out. On the basis of this analysis, the influence of additive and parametric noises is studied. For weak noise, the climate system is localized nearby deterministic attractors. A mixture of the small and large amplitude oscillations caused by noise-induced transitions between equilibria and cycle attraction basins arise with increasing the noise intensity. The portion of large amplitude oscillations is estimated too. The parametric noise introduced in two system parameters demonstrates quite different system dynamics. Namely, the noise introduced in one system parameter increases its dispersion whereas in the other one leads to the stabilization of the climatic system near its unstable equilibrium with transitions from order to chaos.     

Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron

Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure.     

谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌

研究了软弹簧Duffing振子在确定性谐和外力和有界随机噪声联合作用下，系统安全盆的侵蚀和混沌现象.推导出系统的随机Melnikov过程，根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值，然后用数值模拟方法计算了系统的安全盆分叉点.结果表明，由于随机扰动的影响，系统的安全盆分叉点发生了偏移，并且使得混沌容易发生. 关键词： Duffing振子 安全盆 分叉 混沌     

Modification of instability processes by multiplicative noises

We study two dynamical systems submitted to white and Gaussian random noise acting multiplicatively. The first system is an imperfect pitchfork bifurcation with a noisy departure from onset. The second system is a pitchfork bifurcation in which the noise acts multiplicatively on the non-linear term of lowest order. In both cases noise suppresses some solutions that exist in the deterministic regime. Besides, for the first system, the imperfectness of the bifurcation reduces the regime of on-off intermittency. For the second system, the unstable mode can achieve a jump of finite amplitude at instability but without hysteresis. We finally identify a generic property that is verified by the stationary probability density function of the dynamical variable when a control parameter is varied.     

Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps

We demonstrate a method for controlling strong chaos by an aperiodic perturbation in two-dimensional Hamiltonian systems.The method has the advantages that the controlled system remains conservative property and the selection of the perturbation has a considerable diversity.We illustrate this method with two area preserving maps:the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos.Numerical results show that the strong chaos can be effectively controlled into regular motions,and the final states are always quasiperiodic ones.The method is robust against the presence of weak external noise.