Phase synchronization of switching processes in stochastic bistable systems

The concept of an analytic signal is used to show that the stochastic synchronization of bistable systems corresponds to locking of the instantaneous phase of the oscillations in complete agreement with the classical theory of phase synchronization. Pis’ma Zh. Tekh. Fiz. 24, 12–19 (August 12, 1998)     

Phase synchronization of switching processes in a stochastic bistable system with an aperiodic external influence

The synchronization of switching in a stochastic bistable system perturbed by a chaotic signal is analyzed. The effect is described in terms of the classical theory of phase synchronization. Pis’ma Zh. Tekh. Fiz. 24, 14–21 (September 26, 1998)     

Features of induced synchronization of a bistable oscillator with chaotic dynamics

The features of induced synchronization of a Chua oscillator with a nonlinear element admitting cubic approximation have been numerically simulated for a system where the autonomous operation admits regular or chaotic regimes determined by losses in the oscillatory circuit. The cases of deterministic synchronization (under the action of a harmonic external driving signal) and chaotic synchronization (chaotic control) have been considered. It is established that, outside the band of synchronous oscillations that corresponds to the deterministic synchronization, the external harmonic signal stimulates the transformation of regular oscillations into chaos with the motions switched between two attractors. In cases of chaotic synchronization, there is a residual “noise” in the form of differential chaotic oscillations, which grows with increasing non-identity of the driving and driven signals.     

Transient processes during synchronization of coupled bistable oscillators with chaotic dynamics

Transient processes in a chain of nonidentical Chua oscillators resistively coupled via nonlinear elements admitting cubic approximation have been numerically simulated in cases of mutual and unidirectional coupling. It is demonstrated that the transient processes during both mutual and induced (unidirectional) chaotic synchronization are enhanced with increasing detuning between the oscillatory circuits of partial subsystems and suppressed with increasing coefficient of coupling. The time of establishment of a synchronous operation regime in the case of mutual coupling is, under otherwise identical conditions, shorter than in the case of unidirectional coupling. The dependence of the duration of establishment of a synchronous operation regime on the moment of coupling switch-on is considered.     

Synchronization of coupled bistable chaotic systems: experimental study

We carried out an experimental study of the synchronization of two unidirectionally coupled R?ssler-like electronic circuits with two coexisting chaotic attractors. Different stages of synchronization are identified on the route from asynchronous motion to complete synchronization, as the coupling parameter is increased: intermittent asynchronous jumps between coexisting attractors; intermittent anticipating phase synchronization; and generalized synchronization in the form of subharmonic entrainment terminated by complete synchronization. All these regimes are analysed with time-series, power spectra and phase-space plots of the drive and response oscillators. The experimental study implicitly confirms the results of numerical simulations.     

关于非自治Duffing-Van Der Pol 混沌系统完全同步的作用

研究了两类不规则信号——噪声与混沌驱动力在非自治混沌系统同步方面的影响。以谐和力激励的具有 势的Duffing-VanderPol混沌振子为范例,通过对系统的最大Lyapunov指数及同步误差的分析,结果表明：在一定条件下,Gauss白噪声或Rossler混沌驱动力可以诱使两个参数相同的非自治混沌系统达到完全同步;同时通过比较系统的平均同步误差与同步时间,发现噪声诱导的同步对系统的参数失调具有较强的鲁棒性。该研究在生态系统、神经系统及工程等领域具有重要的现实意义和指导意义。     

Some aspects of chaotic and stochastic dynamics for structural systems

In this paper, the bifurcation behaviour of an externally excited four-dimensional nonlinear system is examined. Throughout this paper, a two-degree-of-freedom shallow arch structure under either a periodic or a stochastic excitation will be considered. For the case when the excitation is periodic, the local and global behaviour is examined in the presence of principalsubharmonic resonance and1:2 internal resonance. The method of averaging is used to obtain the first order approximation of the response of the system under resonant conditions. A standard Melnikov type perturbation method is used to show analytically that the system may exhibit chaotic dynamics in the sense of Smale horseshoe for the 1:2 internal resonance case in the absence of dissipation. In the case of stochastic excitation, the stability of the stationary solution is examined by determining themaximal Lyapunov exponent andmoment Lyapunov exponent in terms of system parameters. An asymptotic method is used to obtain explicit expressions for various exponents in the presence of weak dissipation and noise intensity. These quantities provide almost-sure stability boundaries in parameter space. When the system parameters lie outside these boundaries, it is essential to understand the nonlinear behaviour. The method of stochastic averaging is applied to obtain a set of approximate Itô equations which are then examined to describe the local bifurcation behaviour.     

Controlling attractor structure in bistable chaotic oscillator

The effect of the asymmetric characteristic of a nonlinear element in the Chua oscillator on the oscillatory process in this chaotic dynamical system has been studied. The system behavior was studied under the action of irregular oscillations modeling the intrinsic noise, rather than as excited by a single impact. It is established that a growth in the asymmetry parameter of the nonlinear element characteristic leads to an asymmetry of the chaotic attractor of the Chua oscillator that maps the switching of motions between the two basins of attraction. Then, the system exhibits a bifurcation, whereby chaotic oscillations arise in one of these basins and develop into motions that correspond to a multiple limit cycle.     

关于非自治Duffing-Van Der Pol混沌系统完全同步的研究

研究了噪声与混沌驱动力在非自治混沌系统同步方面的影响。以谐和力激励的具有Φ6势的Duffing-Van Der Pol混沌振子为范例,通过对系统的最大Lyapunov指数及同步误差的分析,结果表明:在一定条件下,Gauss白噪声或Rossler混沌驱动力可以诱使两个参数相同的非自治混沌系统达到完全同步;同时通过比较系统的平均同步误差与同步时间,发现噪声诱导的同步对系统的参数失调具有较强的鲁棒性。该研究在生态系统、神经系统及工程等领域具有重要的现实意义和指导意义。     

Generalized synchronization of chaotic oscillators

The behavior of two unidirectionally coupled chaotic oscillators near the boundary of their generalized synchronization is analyzed. Using the modified system method, the position of this boundary on the plane of control parameters is considered and the physical mechanisms leading to the establishment of the generalized synchronization regime are elucidated.     

Duration of the process of complete synchronization of two coupled identical chaotic systems

We consider the time required for complete synchronization of two identical one-way coupled van der Pol-Duffing oscillators occurring in the regime of dynamic chaos. The influence of the initial phase difference between oscillators on the duration of the process of complete synchronization has been studied. At a fixed phase of chaotic oscillations of the self-excited drive oscillator, the period of time (past the coupling onset) during which the complete synchronization regime is established depends on the phase of the self-excited response oscillator.     

Relationship between phase synchronization of chaotic oscillators and time scale synchronization

The relationship between the phase synchronization of chaotic oscillators and their time scale synchronization has been considered, and it is established that the onset of the time scale synchronization depends on the resolving power of the base wavelet. This synchronization usually appears either before or (if said resolving power is insufficient) simultaneously with the onset of phase synchronization. The results are illustrated in the case of one-way coupled Rössler type dynamical systems.     

Deterministic stochastic resonance in a bistable semiconductor system

The phenomenon of deterministic stochastic resonance has been experimentally observed and studied for the first time in a bistable semiconductor system with an S-shaped current-voltage characteristic exhibiting a transition to chaos via intermittency. Spectral regions featuring amplification of an external signal and phase locking are determined as dependent on the frequency and amplitude of the external signal and the control parameter.     

Transformation of bistable chaotic oscillations in triggers of a ring system

Equations describing a ring system of three unidirectionally coupled Chua oscillators (subsystems) are presented. Numerical analysis is performed in the case in which only one subsystem is autooscillatory. It is shown that non-auto-oscillatory subsystems (simplified to single-shot trigger type) exhibit complication of circuit oscillations that is accompanied by multiplication of the oscillation frequencies and expansion of the generated spectrum.     

永磁同步电机系统、网络混沌振荡同步中断分析与控制

已有研究表明,电机在某些参数及工作条件下会出现混沌振荡行为。另一方面,现代工业生产过程中,经常需要多台电机协调同步运行。当前,国内外对电机系统、网络的混沌振荡同步分析与控制已有较多研究,但对电机系统、网络由于参数扰动引起同步中断的分析与控制研究尚属少见。为此,在传统的扩散耦合方式基础上,通过引入单向交叉耦合方式,研究了永磁同步电机(PMSM)系统、网络受到参数扰动后的振动特性。分析并证明了单向交叉耦合对PMSM系统、网络的参数扰动具有同步鲁棒性;并以3节点、4节点和一个多节点网络的PMSM系统验证,交叉耦合对多节点电机网络全局稳定同步的重要作用。研究结果表明,通过建立合适的单向交叉耦合方式,能够有效控制电机网络因参数扰动而引发的同步中断,使同步中断得以重新恢复。     

非线性耦合双稳系统的振动共振与随机共振

单一双稳系统受到弱周期信号和噪声的作用能产生随机共振.而在高、低两种不同频率信号作用下则能产生振动共振,将两个双稳系统经非线性耦合而成为耦合双稳系统,并在低频、高频信号和噪声的作用下研究了耦合双稳系统对低频信号的响应特性,给出了高频信号参数和耦合系数对振动共振和随机共振的影响关系,结果表明通过调节高频信号幅值或耦合系数大小能产生振动共振或随机共振,并构建了耦合双稳系统原理框图,为随机共振的控制及其利用提供了新的理论方法,数值仿真结果与理论分析结论完全吻合.     

Experimental separation of chaotic signals through synchronization

In this paper using a negative feedback scheme we study the problem of synchronizing two systems (each of them made of n independent chaotic circuits) through the transmission of a unique signal (i.e. a scalar variable). To find the appropriate values of the feedback gains, an approach based on the design of an asymptotic observer leading to a set of linear matrix inequalities is used for piecewise linear systems, while for systems with continuous nonlinearities a master stability function approach is adopted. Numerical results showing the suitability of the approach are reported. Furthermore, the experiment showing separation and synchronization of two pairs of chaotic circuits is discussed. Despite the presence of parameter mismatches, separation and synchronization of the two systems can be achieved. This is an experimental demonstration of the successful possibility of multiplexing two (or more) chaotic signals in the same channel.     

Stochastic resonance in a bistable system driven by a chaotic signal

The behavior of a bistable oscillator under the action of a chaotic signal from a Rössler oscillator with a spiral attractor is considered. The influence of the width of the main spectral line of the chaotic drive signal on the signal to noise ratio at the response system’s output has been studied.     

Controlled oscillations in chain of unidirectionally coupled bistable oscillators with chaotic dynamics

Equations of a chain of n (n = 1, 2, ...) unidirectionally coupled bistable oscillators possessing chaotic dynamics are presented. Numerical analysis has been performed for a system of three identical autooscillatory systems with nonlinear elements admitting cubic approximation. It is shown that, as the degree of coupling between units in the chain increases, their motions in the phase space become more complicated and even chaotization of oscillations may take place at the parameters with which separate oscillators exhibit regular motions in the autonomous regimes. The features of the influence of a driving oscillator on the motions of a driven bistable subsystem are related to filtration of the driving signal in the circuit of the driven subsystem.     

Generalized synchronization of chaotic oscillators as a partial case of time scale synchronization

Two coupled dynamical systems of the Rössler type are studied, and it is shown that the generalized synchronization regime can be considered as a partial case of the time scale synchronization process.