The complete synchronization condition in a network of piezoelectric micro-beams

This work deals with the dynamics of a network of piezoelectric micro-beams (a stack of disks). The complete synchronization condition for this class of chaotic nonlinear electromechanical system with nearest-neighbor diffusive coupling is studied. The nonlinearities within the devices studied here are in both the electrical and mechanical components. The investigation is made for the case of a large number of coupled discrete piezoelectric disks. The problem of chaos synchronization is described and converted into the analysis of the stability of the system via its differential equations. We show that the complete synchronization of N identical coupled nonlinear chaotic systems having shift invariant coupling schemes can be calculated from the synchronization of two of them. According to analytical, semi-analytical predictions and numerical calculations, the transition boundaries for chaos synchronization state in the coupled system are determined as a function of the increasing number of oscillators.

     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: Internal resonance activation,reduced-order models and nonlinear normal modes

Resonant multi-modal dynamics due to planar 2:1 internal resonances in the non-linear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1] (in press). The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion.     

Parameter-dependent synchronization of coupled neurons in cold receptor model

In this paper, synchronization in two coupled neurons with spiking, bursting and chaos firings is investigated as the coupling strength gets increased. Synchronization state can be identified by means of the bifurcation diagram, the correlation coefficient and ISI-distance. It is illustrated that the coupled neurons can exhibit different types of synchronization state when the coupling strength increases. The different synchronization processes appear similar, but their detailed processes are different depending on the parameter values. The synchronization of neuronal network with two different network connectivity patterns is also studied. It is shown that chaotic and high period pattern are more difficult to get complete synchronization than the situation in single spike and low period pattern. It is also demonstrated that the synchronization status of multiple neurons is dependent on the network connectivity patterns. These results may be instructive to understand synchronization in neuronal systems.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

Robust chaos synchronizations using an SDRE-based sub-optimal control approach

This paper presents a robust control via a sub-optimal approach, which is achieved by the SDRE control. The proposed approach is to transform a robust control problem into an optimal control problem, where the uncertainties are reflected in the performance index. Thus, the optimal control problem is solved by the SDRE control. Four types of robust control problems are presented, and each type is illustrated by a robust chaos synchronization example. The results show that the proposed approach can effectively regulate the error dynamics of the chaos synchronization.     

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

Synchronization and phase relations in the motion of two-pendulum system

The synchronization phenomenon in non-linear oscillating system is studied by means of examination of the coupled pendulums. Dependence of the phase shift between pendulum states on system parameters and initial conditions is studied both analytically and numerically. The harmonic linearization technique is applied for analytical examinations.     

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.     

Spatiotemporal and synchronous chaos in accumulated coupled map lattice

The onset of spatiotemporal chaos in coupled map lattice (CML) with a new coupling scheme called accumulated CML is studied in this paper. A rigorous proof of the existence of chaos in the sense of Li–Yorke is presented. Also the range of the coupling strength in which global synchronization can be obtained is calculated by stability analysis of the synchronized state. Finally, the positivity of Lyapunov exponents confirms the existence of chaos.     

Nonlinear behaviour of different flexible size-dependent beams models based on the modified couple stress theory. Part 2. Chaotic dynamics of flexible beams

In the present part of the paper various problems of non-linear dynamics of nano-beams within the modified couple stress theory as well as the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson models are studied taking into account the geometric non-linearity. Different characteristics of the vibrational process, including Fourier spectra, wavelet spectra, phase portraits, Poincaré maps as well as the largest Lyapunov exponents, are studied for the same physical-geometric parameter with and without consideration of the size-dependent behaviour. Vibration graphs are constructed and analysed, and scenarios of transition from regular to chaotic vibrations are illustrated and discussed.     

周期激励下广义蔡氏电路混沌运动中的概周期行为

由于广义蔡氏电路存在2个对称的稳定平衡点,周期激励可能导致系统出现相应于不同初值的2种共存的分岔模式. 概周期解由环面破裂进入混沌,混沌吸引子从相位不同步逐渐演化为同步,并进一步随着参数的变化,产生分裂现象. 分裂后的2个相互对称的混沌吸引子仍存在相位同步效应,这2个混沌吸引子再次相互作用后形成扩大了的混沌吸引子,并交替围绕2个子混沌结构来回振荡. 同时,在混沌过程中,其轨迹在相当长的一段时间内严格按照概周期行为振荡,即混沌结构中存在局部概周期行为,这种局部概周期行为随参数的变化会逐步减弱,直至消失.      

Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control

The model and the normalized state equations of the novel version of the Colpitts oscillator designed to operate in the ultra-high frequency range are presented. The circuit is investigated numerically and simulations demonstrate chaos in the microwave frequency range. Typical phase portrait, Lyapunov exponent and Lyapunov dimension are calculated using a piece-wise linear approximation of nonlinear I–V characteristic of the bipolar junction transistor. In addition, the feedback controller is applied to achieve chaos synchronization for two identical improved chaotic Colpitts oscillators. In the frame the nonlinear function of the system is used as a nonlinear feedback term for the stability of the error dynamics. Finally, numerical simulations show that this control method is feasible for this oscillator.     

化学突触耦合神经元的簇放电同步

本文研究了经化学突触耦合的两个神经元的簇放电同步以及耦合后神经元的簇放电动力学性质.根据簇相位的定义,通过计算得到兴奋性耦合导致两个神经元达到同相簇放电同步,而抑制性耦合则使得两个神经元反相同步产生簇放电.本文给出了衡量单个神经元簇动力学的指标-宽度因子,根据此指标将簇放电模式分类为短簇和长簇两种,并且讨论了不同簇放电模式以及耦合方式对于耦合后神经元簇动力学性质的影响.结果表明兴奋性耦合有利于簇放电的整合,短簇的放电模式对于耦合作用具有鲁棒性.这一结果的研究对于将来神经实验中识别簇放电同步具有指导意义.     

Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincaré maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.     

Complexity and emergence of wave dynamics in a chain of sequentially interconnected Chua circuits

This paper considers an ensemble of Chua oscillators bidirectionally coupled in a ring geometry where locally coupled circuits form a closed loop of signal transmission. The spontaneous dynamics of this system is studied numerically for different coupling strength. A transition from periodic to chaotic regimes is observed when the coupling decreases. In the former situation, characterized by high coupling, all the circuits oscillate with pseudo-sinusoidal dynamics on periodic attractors; in the latter they evolve on the same-type of chaotic attractor with a progression of the dynamics from the Chua's spiral to the double scroll as the coupling decreases. The emerging global dynamics is markedly different in the two cases and a phase transition between highly ordered and highly disordered global dynamics is observed. Synchronization and traveling waves moving along the ring are identified in the non-chaotic regime, while spatio-temporal chaos results for very low coupling. Complex patterns formation appears at the “edge of chaos”, for a small couplings interval after the transition between these two regimes.     

Broadband chaos synchronization and communication based on mutually coupled VCSELs subject to a bandwidth-enhanced chaotic signal injection

Based on two mutually coupled vertical-cavity surface-emitting lasers (MC-VCSELs) subject to a bandwidth-enhanced chaotic signal injection, a bidirectional dual-channel broadband chaos communication system is proposed and investigated numerically. The results show that, adopting a bandwidth-enhanced chaotic signal (about 33 GHz) from a driving VCSEL (D-VCSEL) to drive two MC-VCSELs, high-quality isochronal chaos synchronization with over 30 GHz bandwidth between two corresponding LP modes in the two MC-VCSELs can be obtained under proper driving injection, and this synchronization has high tolerance to mismatched intrinsic parameters and frequency detuning. Moreover, based on the broadband chaos synchronization of two corresponding LP modes, the bidirectional dual-channel high-speed chaos communication can be realized and the communication performances have also been preliminarily examined under chaos masking (CMS) encryption scheme.     

Effect of coupling,synchronization of chaos and stick-slip motion in two mutually coupled dynamical systems

In this work, we study the synchronization of two coupled chaotic oscillators. The uncoupled system corresponds to a mass attached to a nonlinear spring and driven by a rolling carpet. For identical oscillators, complete synchronization is analyzed using Lyapunov stability theory. This first analysis reveals that stability area of synchronization increases with the values of the coupling coefficient. Numerical simulations are shown to illustrate and validate stick-slip and chaos synchronization. Some cases of anti-synchronization are detected. Curiously, amplification of fixed point either regular or chaotic is observed in the area of anti-synchronization. Furthermore, phase synchronization is studied for nonidentical oscillators. It appears that for certain values of the coupling coefficient, coincidence of the phases is obtained, while the amplitudes remain uncorrelated. Contrarily to the case of complete synchronization, it does not exist a threshold of the coupling from which phase synchronization could appear. Besides, when we add the modified tuned mass damper on the structure, the behavior of the system can change including the appearance of synchronization, particularly in the region of fixed point. More precisely, complete synchronization is improved in the region of fixed point, while the damage of synchronization is observed when the velocity of the carpets is less than \(0.30\) .     

Diffusive coupling, dissipation, and synchronization

We consider the synchronization of diffusive coupled systems in situations where the synchronization is a consequence of the dissipation in the coupling as well as ones where there is an interaction between the inherent damping in the subsystems and the coupling. It is not required that the subsystems be identical, and they are allowed to have chaotic dynamics. Both discrete and continuous versions are discussed. We also consider coupled oscillators where the dynamics of each oscillator is determined by circuitry across a lossless transmission line.