Pure odd-order oscillators with constant excitation

In this paper the excited vibrations of a truly nonlinear oscillator are analyzed. The excitation is assumed to be constant and the nonlinearity is pure (without a linear term). The mathematical model is a second-order nonhomogeneous differential equation with strong nonlinear term. Using the first integral, the exact value of period of vibration i.e., angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained. The closed form solution has the form of gamma function. The period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term. For the case of pure odd-order-oscillators the approximate solution of differential equation is obtained in the form of trigonometric function. The solution is based on the exact value of period of vibration. For the case when additional small perturbation of the pure oscillator acts, the so called ‘Cveticanin's averaging method’ for a truly nonlinear oscillator is applied. Two special cases are considered: one, when the additional term is a function of distance, and the second, when damping acts. To prove the correctness of the method the obtained results are compared with those for the linear oscillator. Example of pure cubic oscillator with constant excitation and linear damping is widely discussed. Comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement. The investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems. For example, such mechanical systems are seats in vehicles, supports for machines, cutting machines with periodical motion of the cutting tools, presses, etc. The examples can be find in electronics (electromechanical devices like micro-actuators and micro oscillators), in music instruments (hammers in piano), in human voice producing folds (voice cords), etc.     

Periodically kicked hard oscillators

A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval; the topological structure of this object is generated by the finite size of the repelling set, and is therefore also intrinsic to hard oscillators.     

Hydrodynamic synchronization of flagellar oscillators

In this review, we highlight the physics of synchronization in collections of beating cilia and flagella. We survey the nonlinear dynamics of synchronization in collections of noisy oscillators. This framework is applied to flagellar synchronization by hydrodynamic interactions. The time-reversibility of hydrodynamics at low Reynolds numbers requires swimming strokes that break time-reversal symmetry to facilitate hydrodynamic synchronization. We discuss different physical mechanisms for flagellar synchronization, which break this symmetry in different ways.     

Mutual synchronization of molecular oscillators

Radiophysics and Quantum Electronics -     

Complete synchronization of Kuramoto oscillators with finite inertia

We present an approach based on Gronwall’s inequalities for the asymptotic complete phase-frequency synchronization of Kuramoto oscillators with finite inertia. For given finite inertia and coupling strength, we present admissible classes of initial configurations and natural frequency distributions, which lead to the complete phase-frequency synchronization asymptotically. For this, we explicitly identify invariant regions for the Kuramoto flow, and derive second-order Gronwall’s inequalities for the evolution of phase and frequency diameters. Our detailed time-decay estimates for phase and frequency diameters are independent of the number of oscillators. We also compare our analytical results with numerical simulations.     

Recurrence analysis and synchronization of oscillators with coexisting attractors

The method of recurrence plots (RPs) has been traditionally used for experimental time series analysis with no comparison with the mathematical model. This is in part because of lack of nonlinear analysis of mathematical model based on the recurrence quantification analysis (RQA) parameters. The paper provides substantial information about the mathematical and numerical analysis and synchronization of a multi-limit cycle oscillator from the RQA perspective. The recurrence quantification analysis parameters are used to discuss the birhythmic behavior of the system, as well as various bifurcations (quasi-periodicity, periodicity and chaos) in the system response. Finally, the results of the method of RPs are compared to those of phase diagrams and the problem of synchronization of limit cycle and chaotic response is discussed by the mean of cross recurrence.     

Subfemtosecond synchronization of microwave oscillators with mode-locked Er-fiber lasers

We synchronize an 8.06 GHz microwave signal from a voltage-controlled oscillator with an optical pulse train from a 77.5 MHz mode-locked Er-fiber laser using a fiber-based optical-microwave phase detector. The residual phase noise between the optical pulse train and the synchronized microwave signal is -133 dBc/Hz (-154 dBc/Hz) at 1 Hz (5 kHz) offset frequency, which results in 838 as integrated rms timing jitter [1 Hz-1 MHz]. The long-term residual phase drift is 847 as (rms) measured over 2 h, which reaches 4×10(-19) fractional frequency instability at 1800 s averaging time. This method has a potential to provide both subfemtosecond-level short-term phase noise and long-term phase stability in microwave extraction from mode-locked fiber lasers.     

Explosive synchronization of complex networks with different chaotic oscillators

Recent studies have shown that explosive synchronization transitions can be observed in networks of phase oscillators [Gómez-Garden es J,Gómez S,Arenas A and Moreno Y 2011 Phys.Rev.Lett.106 128701] and chaotic oscillators [Leyva I,Sevilla-Escoboza R,BuldúJ M,Sendin a-Nadal I,Gómez-Garden es J,Arenas A,Moreno Y,Gómez S,Jaimes-Reátegui R and Boccaletti S 2012 Phys.Rev.Lett.108 168702].Here,we study the effect of different chaotic dynamics on the synchronization transitions in small world networks and scale free networks.The continuous transition is discovered for Rssler systems in both of the above complex networks.However,explosive transitions take place for the coupled Lorenz systems,and the main reason is the abrupt change of dynamics before achieving complete synchronization.Our results show that the explosive synchronization transitions are accompanied by the change of system dynamics.     

Noise-enhanced phase synchronization of chaotic oscillators

The effects of noise on phase synchronization (PS) of coupled chaotic oscillators are explored. In contrast to coupled periodic oscillators, noise is found to enhance phase synchronization significantly below the threshold of PS. This constructive role of noise has been verified experimentally with chaotic electrochemical oscillators of the electrodissolution of Ni in sulfuric acid solution.     

超混沌振荡器的混沌耦合同步

提出了两种应用单变量单向混沌耦合同步超混沌振荡器的同步方案。首先给出了混沌耦合同步方案的数学证明，然后通过数值仿真证实了该混沌耦合同步方案的正确性和有效性。最后通过计算条件李雅谱诺夫指数，比较了不同耦合强度下，两种混沌耦合同步和典型的连续耦合同步的同步性能。计算表明在一定的耦合强度下，混沌耦合比典型的连续耦合同步速度快。     

Noise-induced phase synchronization and synchronization transitions in chaotic oscillators

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.     

Elastic-plastic oscillators under random excitation

Simulation studies reveal that the elastic-plastic oscillator cannot be successfully treated by any linearization procedure. In order to compute the main response quantities, such as the yielding increment, the permanent set, the dissipated energy and the crossing rates, a so-called two-state approach is suggested. With the separation between the elastic and plastic parts of the response kept as originally proposed by Karnopp and Scharton, one linear equation is obtained for each state. The fundamental step then is to set up a conditional probability density for the plastic state variable under Gaussian stationary or non-stationary white noise excitation. The yielding duration is found to be half-normal distributed. Numerical results demonstrate the approach and the range of parameters where it can be successfully applied.     

Phase synchronization and synchronization frequency of two-coupled van der Pol oscillators with delayed coupling

In this paper, phase synchronization and the frequency of two synchronized van der Pol oscillators with delay coupling are studied. The dynamics of such a system are obtained using the describing function method, and the necessary conditions for phase synchronization are also achieved. Finding the vicinity of the synchronization frequency is the major advantage of the describing function method over other traditional methods. The equations obtained based on this method justify the phenomenon of the synchronization of coupled oscillators on a frequency either higher, between, or lower than the highest, in between, or lowest natural frequency of the aggregate oscillators. Several numerical examples simulate the different cases versus the various synchronization frequency delays.     

Engineering generalized synchronization in chaotic oscillators

We report a method of engineering generalized synchronization (GS) in chaotic oscillators using an open-plus-closed-loop coupling strategy. The coupling is defined in terms of a transformation matrix that maps a chaotic driver onto a response oscillator where the elements of the matrix can be arbitrarily chosen, and thereby allows a precise control of the GS state. We elaborate the scheme with several examples of transformation matrices. The elements of the transformation matrix are chosen as constants, time varying function, state variables of the driver, and state variables of another chaotic oscillator. Numerical results of GS in mismatched Ro?ssler oscillators as well as nonidentical oscillators such as Ro?ssler and Chen oscillators are presented.     

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     

Explosive synchronization in network of mobile oscillators

Recently, the explosive synchronization (ES) has attracted great interests. Motivated by the recent dynamic framework of complex network, we focus on the network of mobile oscillators and study synchronization phenomenon. The local synchronous order parameter of the neighbors of the oscillator is used as the controllable variable to adjust the coupling strength of the oscillator. Hence, it can be seen as a kind of adaptive strategy. By numerical simulation, we find that ES can be observed in the dynamic network of mobile oscillators, accompanying with hysteresis loop, as the coupling strength increases gradually. It is found that the critical value of coupling strength and hysteresis loop width is affected by the natural frequency distribution and the number of neighbors the oscillator owning. It can be deduced that ES will be motivated by increasing the number of oscillators in the network. Meanwhile, our results are feasible to different natural frequency distributions, such as Lorentzian, Gaussian power-law, and Rayleigh distribution, whether it is symmetric or not.     

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。     

Phase equation with nonlinear excitation for nonlocally coupled oscillators

Some reaction-diffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is effectively a diffusion equation which is excitation-free. However, more complex forms are possible such as the Nikolaevskii equation and the Kuramoto–Sivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.