Phase synchronization in unidirectionally coupled chaotic ratchets

We study chaotic phase synchronization of unidirectionally coupled deterministic chaotic ratchets. The coupled ratchets were simulated in their chaotic states and perfect phase locking was observed as the coupling was gradually increased. We identified the region of phase synchronization for the ratchets and show that the transition to chaotic phase synchronization is via an interior crisis transition to strange attractor in the phase space.     

Synchronization of linearly coupled networks of deterministic ratchets

This Letter focuses on the synchronization in a class of dynamical complex networks with each node being a deterministic ratchet. In virtue of the technique derived from pendulum-like nonlinear analytic theory and Kalman–Yakubovich–Popov (KYP) lemma, simple linear matrix inequality (LMI) formulations are established to guarantee the stable synchronization of such networks. An interesting conclusion is reached that the stability of synchronization in the coupled whole N-dimensional networks can be converted into that of the simplest 2-dimensional space.     

Complete synchronization between two bi-directionally coupled chaotic systems via an adaptive feedback controller

In this paper, we apply a simple adaptive feedback control scheme to synchronize two bi-directionally coupled chaotic systems. Based on the invariance principle of differential equations, sufficient conditions for the global asymptotic synchronization between two bi-directionally coupled chaotic systems via an adaptive feedback controller are given. Unlike other control schemes for bi-directionally coupled systems, this scheme is very simple to implement in practice and need not consider coupling terms. As examples, the autonomous hyperchaotic Chen systems and the new non-autonomous 4D systems are illustrated. Numerical simulations show that the proposed method is effective and robust against the effect of weak noise.     

Transition to complete synchronization via near-synchronization in two coupled chaotic neurons

The synchronization transition in two coupled chaotic Morris-Lecar （ML） neurons with gap junction is studied with the coupling strength increasing. The conditional Lyapunov exponents, along with the synchronization errors are calculated to diagnose synchronization of two coupled chaotic ML neurons. As a result, it is shown that the increase in the coupling strength leads to incoherence, then induces a transition process consisting of three different synchronization states in succession, namely, burst synchronization, near-synchronization and embedded burst synchronization, and achieves complete synchronization of two coupled neurons finally. These sequential transitions to synchronization reveal a new transition route from incoherence to complete synchronization in coupled systems with multi-time scales.     

Deterministic inhomogeneous inertia ratchets

We study the deterministic dynamics of a periodically driven particle in the underdamped case in a spatially symmetric periodic potential. The system is subjected to a space-dependent friction coefficient, which is similarly periodic as the potential but with a phase difference. We observe that frictional inhomogeneity in a symmetric periodic potential mimics most of the qualitative features of deterministic dynamics in a homogeneous system with an asymmetric periodic potential. We point out the need of averaging over the initial phase of the external drive at small frictional inhomogeneity parameter values or analogously low potential asymmetry regimes in obtaining ratchet current. We also show that at low amplitudes of the drive, where ratchet current is not possible in the deterministic case, noise plays a significant role in realizing ratchet current.     

Controlling projective synchronization in coupled chaotic systems

In this paper, a new method for controlling projective synchronization in coupled chaotic systems is presented. The control method is based on a partially linear decomposition and negative feedback of state errors. Firstly, the synchronizability of the proposed projective synchronization control method is proved mathematically. Then, three different representative examples are discussed to verify the correctness and effectiveness of the proposed control method.     

Impulsive synchronization and control of directed transport in chaotic ratchets

The impulsive synchronization problem of two identical chaotic ratchets is investigated in this paper. We demonstrate that the impulsive method to control directed transport is applicable when there are multiple co-existing attractors in phase space transporting particles in different directions. Numerical simulations are carried out to illustrate the effectiveness of the proposed method.     

强耦合混沌系统中的近似同步

以单向驱动耦合Lorenz振子一维链为研究对象，研究振子间的混沌同步行为. 数值计算结果表明，对于变量y驱动x的耦合方式，在合适的耦合强度下，会出现第一个振子和第二个振子不同步，而与次近邻非直接连接的振子(如第三个振子)近似同步. 进一步研究表明，出现这一现象的原因是在大耦合强度下，对于这种驱动方式，第一个振子和第二个振子间出现驱动单变量近似同步；虽然它们之间未出现所有变量的完全同步，但是驱动信号事实上已经传递下去了. 关键词： Lorenz振子 混沌同步 近似同步     

双频驱动混沌系统的相同步和广义同步

研究了双频混沌信号驱动的混沌振子的广义同步和相同步问题.发现了反偏向的相同步和正偏向的广义同步，即响应振子可以优先与驱动强度弱的混沌信号达到相同步，而广义同步则先在驱动强的信号和响应振子间建立起来.对这些行为产生的动力学机理进行了详细地分析. 关键词： 相同步 广义同步 条件熵 平均频率     

Return map analysis of chaotic phase synchronization

We investigate chaotic phase synchronization (CPS) observed in a mutually coupled map system and a forced map oscillator by using return maps (RM) of phase difference. RM visualizes the presence or absence of the channel for phase slip which characterizes the onset of CPS. It is found that CPS occurs when the channel starts to be completely closed, and that the saddle node point is approximately equal to the point at which the channel is closed on the average. Furthermore, the proposed RM describes the critical characteristics of the phase difference and reveals the existence of the quasi-synchronized state below the synchronization point.     

Phase synchronization in fractional differential chaotic systems

We propose an analytical justification for phase synchronization of fractional differential equations. This justification is based upon a linear stability criterion for fractional differential equations. We then investigate the existence of phase synchronization in chaotic forced Duffing and Sprott-L fractional differential systems of equations. Our numerical results agree with those analytical justifications.     

Phase synchronization in two coupled chaotic neurons

Chaotically-spiking dynamics of Hindmarsh–Rose neurons are discussed based on a flexible definition of phase for chaotic flow. The phase synchronization of two coupled chaotic neurons is in fact the spike synchronization. As a multiple time-scale model, the coupled HR neurons have quite different behaviors from the Rössler oscillators only having single time-scale mechanism. Using such a multiple time-scale model, the phase function can detect synchronization dynamics that cannot be distinguished by cross-correlation. Moreover, simulation results show that the Lyapunov exponents cannot be used as a definite criterion for the occurrence of chaotic phase synchronization for such a system. Evaluation of the phase function shows its utility in analyzing nonlinear neural systems.     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

基于线性耦合下混沌系统的同步条件

将混沌同步问题用精确的数学语言给予描述，通过数学分析将其转化为微分方程组的稳定性问题.以Lorenz系统族的Chen氏系统作为典型系统，在线性耦合下分析系统参数，得到了系统达到同步时的充分条件，且在理论上加以证明.结合该条件，提出了一种确定耦合系数的方法.最后用仿真实验验证了该方法的正确性，并验证了在不同耦合方式和参考系统的情况下定理的有效性. 关键词： 混沌同步 耦合 稳定性     

Phase synchronization in tilted deterministic ratchets

We study phase synchronization for a ratchet system. We consider the deterministic dynamics of a particle in a tilted ratchet potential with an external periodic forcing, in the overdamped case. The ratchet potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of external periodic forcing. This oscillator has an intrinsic frequency that can be entrained with the frequency of the external driving. We introduced a linear phase through a set of discrete time events and the associated average frequency, and show that this frequency can be synchronized with the frequency of the external driving. In this way, we can properly characterize the phenomenon of synchronization through Arnold tongues, which represent regions of synchronization in parameter space, and discuss their implications for transport in ratchets.     

Complete synchronization induced by disorder in coupled chaotic lattices

The effect of phase disorder in external forces introduced into two-dimensional lattices of coupled chaotic pendulums is investigated. As the increase of the disorder, we find complete synchronization between the pendulums in each chain and different periodic synchronized patterns, while the chain remains asynchronous if all driving forces have the same phase. Applying the master stability function method, an analytic solution is given to support the numerical results. All these findings may provide further insight into chaos control and synchronization in nonlinear systems.     

Projective synchronization in coupled fractional order chaotic Rossler system and its control

This paper proposes a method to achieve projective synchronization of the fractional order chaotic Rossler system. First, construct the fractional order Rossler system's corresponding approximate integer order system, then a control method based on a partially linear decomposition and negative feedback of state errors is utilized on the new integer order system. Mathematic analyses prove the feasibility and the numerical simulations show the effectiveness of the proposed method.     

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.     

Study on phase synchronization of stochastic chaotic system

This paper detects and characterizes the diverse roles played by bounded noise in chaotic phase synchronization (CPS) of weakly coupled nonlinear stochastic systems. Analysis of a paradigmatic model of two bidirectional coupled three-level food chains is carried out by various statistical measures such as Shannon entropy and mutual information. The results indicate that inside the synchronous regime, CPS is considerably reduced under the influence of bounded noise; near the onset of phase synchronization, temporal phase locking is diversely changed with the increase of noise, i.e., either weak or strong noise also degrades the degree of CPS, while intermediate noise enhances CPS remarkably, and an optimal noise intensity is detected that maximizes the enhancement.     

Modified linear-nonlinear decomposition method for chaotic synchronization

This Letter proposes a general linear-nonlinear decomposition method for chaotic synchronization. The method expands the concept of chaotic synchronization based on the stability criterion of linear systems, proposed earlier. Retaining the basic idea of the standard linear-nonlinear decomposition method—the nonlinear part of a given chaotic system is used as a synchronization signal, hence the error system is always linear and allows precise stability analysis—the proposed method allows to design not only one, but a large number of coupling variants, thus offering the researcher the possibility to choose the best possible coupling variant for a given chaotic synchronization problem.