Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincar map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using non-feedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to period-motions by adding an excitation term.     

准周期激励与应变超晶格的动力学稳定性

假设超晶格锯齿形沟道对粒子的作用等效为形状相似的周期场作用. 在经典力学框架内,引入正弦平方势,把粒子运动方程化为具有阻尼项和双频激励项的摆方程. 用Melnikov方法对单频激励系统的分叉与混沌进行分析;用Lyapunov方法对双频激励系统的稳定性进行讨论. 结果表明：在弱非线性情况下,双频激励系统存在局域不稳定,且这种不稳定将向全局扩展,直至混沌的出现;导致混沌的双频激励强度远小于单频激励强度;外加一个适当的超声场可望将这种敏感钝化,使系统的稳定性得到改善. 关键词： 超晶格 准周期激励 混沌 稳定性     

一类含Mathieu-Duffing振子的相对转动系统的分岔和混沌

建立了一类具有Mathieu-Duffing振子的两质量相对转动系统的非线性动力学方程. 应用多尺度法求解该系统发生主共振-基本参数共振的分岔响应方程,并通过奇异性分析得到系统稳态响应的转迁集. 利用Melnikov方法讨论系统在外激扰动和参激扰动变化下的全局分岔和系统进入混沌状态的可能途径,得到外激和参激幅值变化下系统可能出现多次通向混沌的道路,获得系统发生混沌的必要条件. 最后采用数值方法验证了理论研究的有效性. 关键词： 相对转动 Mathieu-Duffing振子 混沌 Melnikov方法     

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.     

一类约瑟夫森结混沌系统的谐和共振控制

引入外激和参激两种不同形式的谐和共振激励,探讨了一类约瑟夫森结(Josephson junction)系统的混沌控制问题.利用Melnikov方法研究了异宿混沌的生成和抑制,得到了在一定的控制激励振幅范围内,能确保异宿混沌被控制住,而且推导出控制激励与系统的激励两者之间的相位差和两者频率之间的共振阶数应满足的关系式.从定性的角度说明相位差在异宿混沌的控制中确实有着至关重要的影响,而且,数值方法的研究表明可通过调节相位来控制非自治系统中的稳态混沌.通过分析、比较外激和参激两种不同的共振激励对约瑟夫森结系统的异宿混沌的控制效果,得到对于较小的共振频率,宜采用参激激励,而对于较大的共振频率,宜采用外激激励. 关键词： 混沌控制 谐和共振激励 相位控制 Melnikov方法     

一类非线性相对转动系统的混沌运动及多时滞反馈控制

建立一类含非线性粘滑摩擦力的两质量非线性相对转动系统的动力学方程. 研究此非线性相对转动系统在外激励作用下的混沌运动及多时滞反馈控制. 当系统在外激励作用下处于混沌状态时, 考虑引入多时滞反馈对系统的混沌运动进行控制. 应用Melnikov理论给出系统在Smale意义下的混沌临界条件, 研究了多时滞反馈对系统运动及混沌临界值的影响规律. 并结合系统相图、Poincare截面图和功率谱分析多时滞反馈参数对系统混沌运动的控制作用. 关键词： 多时滞 相对转动 控制 数值仿真     

Suppression of chaos in the vicinity of a separatrix

The standard Melnikov method for analyzing the onset of chaos in the vicinity of a separatrix is used to explore the possibility of suppressing chaos of dynamical systems of a certain class. Analytical expressions are obtained for external perturbations that eliminate chaotic behavior. These results are supplemented with a numerical analysis of the Duffing-Holmes-oscillator and pendulum equations.     

Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection

The problem of suppressing chaos in the Rayleigh oscillator with fractional order deflection is considered. The explanation of Melnikov?s techniques shows that the dynamic performance and robustness of the system are highly dependent on the fractional order α. The feedback control system is considered as active control strategy. It is revealed with analytical results that periodic perturbation from the controller enhances the performance of the active control strategy. The proposed control strategy is more efficient for deflection order α∈[1.5,2.5] and under super resonant condition between the driven frequency and perturbation frequency. Numerical simulations demonstrate the effectiveness of Melnikov?s analysis.     

Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise

The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied.The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.     

用低通滤波方法终止心脏组织中的螺旋波和时空混沌

通过让心肌细胞钠离子通道的触发门变量延迟打开, 使介质具有激发延迟能力, 介质延迟激发时间随控制电压和刺激频率增加而增加, 当控制电压超过一个阈值时, 延迟激发介质具有低通滤波作用:低频波可以连续通过, 而高频波不能连续通过. 本文用Luo-Rudy相I模型研究了介质延迟激发对螺旋波和时空混沌的影响, 数值模拟结果表明: 当控制电压超过阈值时, 介质的延迟激发可有效消除螺旋波和时空混沌; 从小逐渐增大控制电压, 在钙最大电导率较小情况下, 延迟激发会导致介质激发性降低, 使螺旋波漫游幅度增大, 直至传导障碍导致螺旋波消失; 当钙最大电导率较大时, 延迟激发会导致螺旋波失稳变弱, 这样当控制电压增加到一定值时, 时空混沌可以演化成漫游螺旋波, 当控制参数被适当选取时, 观察到漫游幅度大的螺旋波漫游出系统边界消失现象, 继续增大控制电压将导致时空混沌直接消失.     

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     

Melnikov's criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation

We consider the problems of chaos and parametric control in nonlinear systems under an asymmetric potential subjected to a multiscale type excitation. The lower bound line for horseshoes chaos is analyzed using the Melnikov's criterion for a transition to permanent or transient nonperiodic motions, complement by the fractal or regular shape of the basin of attraction. Numerical simulations based on the basins of attraction, bifurcation diagrams, Poincare? sections, Lyapunov exponents, and phase portraits are used to show how stationary dissipative chaos occurs in the system. Our attention is focussed on the effects of the asymmetric potential term and the driven frequency. It is shown that the threshold amplitude ∣γ(c)∣ of the excitation decreases for small values of the driven frequency ω and increases for large values of ω. This threshold value decreases with the asymmetric parameter α and becomes constant for sufficiently large values of α. γ(c) has its maximum value for asymmetric load in comparison with the symmetric load. Finally, we apply the Melnikov theorem to the controlled system to explore the gain control parameter dependencies.     

Duffing系统的主-超谐联合共振

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到稳态周期解的稳定性条件,并分析了非线性刚度对稳态周期解的幅值和稳定性的影响.此外,由于近似解只能描述周期运动,不足以描述系统的全局特性,因而应用Melnikov方法对系统进行全局分析,得到系统进入Smale马蹄意义下混沌的条件,依据该条件以及主-超谐联合共振的条件选取一组参数进行数值仿真.分岔图和最大Lyapunov指数显示出两个临界值:当激励幅值通过第一个临界值时,异宿轨道破裂,混沌吸引子突然出现,系统以激变方式进入混沌;激励幅值通过第二个临界值时,系统在混沌态下再次发生激变,进入另一种混沌态.利用Melnikov方法考察了第一个临界值在多种频率组合下的变化趋势,并用数值仿真验证了解析结果的正确性.     

Sliding Mode Control of Fractional-Order Delayed Memristive Chaotic System with Uncertainty and Disturbance

In this paper, a simplest fractional-order delayed memristive chaotic system is proposed in order to control the chaos behaviors via sliding mode control strategy. Firstly, we design a sliding mode control strategy for the fractionalorder system with time delay to make the states of the system asymptotically stable. Then, we obtain theoretical analysis results of the control method using Lyapunov stability theorem which guarantees the asymptotic stability of the noncommensurate order and commensurate order system with and without uncertainty and an external disturbance. Finally,numerical simulations are given to verify that the proposed sliding mode control method can eliminate chaos and stabilize the fractional-order delayed memristive system in a finite time.     

Stabilizing Chaos in the rf-Driven Josephson Junction

The models of rf-driven Josephson junctions are investigated as the perturbed systems.It is shown that the heteroclinic orbits corrected by perturbations are stable if some relations between the initial constants and system parameters are satisfied, which leads to Melnikov chaos. To stabilize chaos one has to make the control parameters fitting the relations.The result is compared with the previous numerical work and a good agreement is found.     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

待定固有频率法在分析系统混沌临界值问题中的应用

利用规范形理论与待定固有频率寻求改善Melnikov函数分析非线性振动系统混沌阈值的简单方法,着重讨论参数与周期激励联合作用下具有主参数共振的三阱势能系统,建立了其Melnikov函数积分式.引入由待定固有频率形成的时间尺度变换,从同宿及异宿分岔两个角度获取系统的混沌临界值,使得非线性扰动量对于基频的影响有效地体现于Melnikov函数表达式中,进而结合相应的分析过程提高所得结果的计算精度.作为算例,对解析解与数值积分结果进行了对比,以验证提出方法的有效性与可行性. 关键词： 规范形 Melnikov方法 混沌 同宿分岔     

Chaotic behaviour from smooth and non-smooth optical solitons under external perturbation

Smooth and non-smooth optical solitons in the nonlinearly dispersive Schrödinger equation are given by phase portraits. The Melnikov technique is used to detect conditions for chaotic motion of this deterministic system and to analyse conditions for the suppression of chaos. Our results show that the system is in a state of Melnikov chaos by external disturbances. After the implementation of the controlled system, the optical solitons can transmit in a stable station for a long time. Numerical simulation also shows that maximum interference frequency of the system enables the dynamic behaviour to be more complex. The effect of controller parameter on phase portraits as well as on the numerical simulations of bifurcation diagram and maximum Lyapunov exponents are also investigated.     

Dynamical behaviors of the shock compacton in the nonlinearly Schrödinger equation with a source term

In this paper, the dynamics from the shock compacton to chaos in the nonlinearly Schrödinger equation with a source term is investigated in detail. The existence of unclosed homoclinic orbits which are not connected with the saddle point indicates that the system has a discontinuous fiber solution which is a shock compacton. We prove that the shock compacton is a weak solution. The Melnikov technique is used to detect the conditions for the occurrence from the shock compacton to chaos and further analysis of the conditions for chaos suppression. The results show that the system turns to chaos easily under external disturbances. The critical parameter values for chaos appearing are obtained analytically and numerically using the Lyapunov exponents and the bifurcation diagrams.     

Effect of rectified and modulated sine forces on chaos in Duffing oscillator

The effect of rectified and modulated sine forces on the onset of horseshoe chaos is studied both analytically and numerically in the Duffing oscillator. With single force analytical threshold condition for the onset of horseshoe chaos is obtained using the Melnikov method. The Melnikov threshold curve is drawn in a parameter space. For the rectified sine wave, onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of chaos by the addition of second modulated force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted analytically and verified numerically. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.