两自由度微扰力学系统的二阶近似守恒量

把微扰力学系统视为未受微扰系统与微扰项的迭加,并选择合适的方法求得未受微扰系统的精确守恒量I0.从近似守恒量的性质出发,建立守恒量的一阶微扰项系数I1与精确守恒量I0、守恒量的二阶微扰项系数I2与守恒量的一阶微扰项系数I1及精确守恒量I0的递推关系.考虑微扰项对精确守恒量以及对守恒量的一阶微扰项系数的影响,利用递推关系并直接积分求得二阶近似守恒量.文中用此方法研究了一微扰力学系统的二阶近似守恒量,并得到2个稳定的二阶近似守恒量.     

微扰Kepler系统的守恒量与对称性

用直接积分法和Noether法研究微扰Kepler系统的守恒量,都得到了一个不同于Hamilton函数的守恒量,此守恒量与Runge—Lenz矢量有相同的量纲,可以称其为“类Runge-Lenz矢量守恒量”．文中还讨论了守恒量的Noether对称性、Lie对称性与Mei对称性,结果表明：与守恒量相应的无限小变换同时是Noether对称变换、Lie对称变换和Mei对称变换．     

Van der Pol振动系统同步时间与反馈增益的关系

在受迫Van der Pol振动系统的近似解的基础上,获得驱动系统的虚拟轨线.将虚拟轨线代入驱动-响应振动系统的近似误差方程,再用多尺度法求得同步时间关于反馈增益的分析表达式,并且将数值与分析结果进行比较表明:用该方法求得的同步时间与反馈增益的关系和数值模拟结果相当一致.这方法也适用于研究自激Van der Pol振动系统.     

分数阶奇异系统的Lie对称性与守恒量

对称性与守恒量可以简化动力学问题从而进一步求出力学系统的精确解,这样更加有利于研究动力学行为.分数阶模型相比于整数阶模型,能够描述复杂系统的动力学过程,因此在分数阶模型下研究对称性与守恒量是不可或缺的.首先介绍两个分数阶奇异系统,一个系统包含混合整数和Caputo分数阶导数,另一个系统仅含Caputo分数阶导数.由两个分数阶奇异系统分别给出两个分数阶固有约束,并给出对应的分数阶约束Hamilton方程.然后,基于微分方程在无限小变换下的不变性,给出了分数阶约束Hamilton方程Lie对称性的定义,导出了相应的确定方程,限制方程和附加限制方程.第三,建立并证明了两个分数阶约束Hamilton系统的Lie对称性定理,得到了相应的分数阶约束Hamilton系统的Lie守恒量.在特定条件下,本文所得结果可以退化为整数阶约束Hamilton系统的Lie守恒量.最后通过两个算例来说明此结果的应用.     

多自由度粘弹性非线性随机系统的瞬态响应

研究了高斯白噪声激励下多自由度粘弹性非线性系统的瞬态响应.首先,通过将粘弹性项对系统的作用近似地简化为对原系统阻尼部分以及刚度部分的修正,得到近似的不具粘弹性项的等效非线性随机系统.然后,应用基于广义谐和函数的随机平均法,导出关于幅值瞬态概率密度的平均Fokker-Planck-Kolmogorov方程.该方程的解可通过多重级数式表示,基函数为幅值相关正交函数,系数为时间函数.应用Galerkin方法,关于时间的系数可由一阶线性微分方程组解得,从而得出幅值响应的瞬态概率密度、状态空间概率密度及幅值统计矩的半解析表达式.最后,以耦合的二自由度Duffing-van der Pol振子系统为例,通过与原系统数值模拟结果的比较分析验证了所提出的半解析方法的有效性,并讨论了粘弹性对系统响应的影响.     

一类强非线性系统共振周期解的渐近分析

强非线性系统经引入参数变换,并在一定的假设条件下,可转化为弱非线性系统．将其解展成为改进的傅立叶级数后,利用参数待定法可方便地求出强非线性系统的共振周期解．研究了Duffing方程的主共振、Van der Pol方程的3次超谐共振和Van der Pol-Mathieu方程的1／2亚谐共振周期解．这些例子表明近似解与数值解非常吻合。     

分数阶Birkhoff系统基于Caputo导数的Noether对称性与守恒量

在Caputo分数阶导数下研究分数阶Birkhoff系统的Noether对称性与守恒量.首先,定义Caputo分数阶导数下的分数阶Pfaff作用量,建立分数阶Birkhoff方程及其相应的横截性条件;其次,基于Pfaff作用量在无限小变换下的不变性,分别在时间不变和时间变化的无限小变换下,给出了不变性条件.基于Frederico和Torres的分数阶守恒量概念,建立了分数阶Birkhoff系统的Noether定理,揭示了分数阶Noether对称性与分数阶守恒量之间的内在联系.     

Chetaev型非完整系统Nielsen方程Lie对称性导致的一种守恒量

研究Chetaev型非完整系统Nielsen方程Lie对称性导致的一种守恒量,给出无限小群变换下Chetaev型非完整系统Nielsen方程Lie对称性的确定方程,得到Chetaev型非完整系统Nielsen方程Lie对称性直接导致的一种守恒量及其存在条件,并举例说明结果应用.     

平面二级倒立摆的圆周行走与镇定控制

采用拉格朗日方程建立了平面二级倒立摆的非线性动力学模型, 将其在平衡位置线性化, 得到系统在两个正交控制方向解耦的近似模型. 针对每一个方向上由互相耦合的基座小车定位子系统和摆杆镇定子系统所组成的六阶欠驱动系统, 设计了自适应滑模模糊控制器, 实现了基座小车沿圆周行走条件下摆杆的镇定控制, 验证了该控制算法对欠驱动、不稳定、多变量耦合系统的有效性.     

异结构不确定时变时滞复杂网络的外同步

本文研究了具有时间延迟的异结构不确定网络之间的外同步问题.基于Lyapunov稳定性理论,采用双向耦合自适应方法实现了两个结构互异的复杂网络之间的外同步.并且,网络的拓扑结构以及耦合强度也被同时确定.在数值模拟中,选取Van der Pol系统和Duffing系统作为网络节点进行仿真模拟,验证其理论结果的有效性.     

Approximate solutions to Van der Pol damped nonlinear oscillators by means of He's energy balance method

In this article, He's energy balance method (EBM) is applied to solve Van der Pol damped nonlinear oscillators. Three examples of Van der Pol oscillators are presented and solved by this method to illustrate the effectiveness and convenience of the EBM. In this method, only one iteration leads to high accuracy of the solution. Comparisons are made between EBM results and exact solutions of the problems. The results show that the method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.     

Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method

Although the Van der Pol oscillator, which was originally proposed as a model of vacuum tube circuits, has been widely used in electronics, biology and acoustics, its characteristics in fractional order formulations are not clearly explained even now. This paper is interested in gaining insights of approximate expressions of the periodic solutions in a fractional order Van der Pol oscillator. The presence of fractional derivatives requires the use of suitable criteria, which usually makes the analytical work much hard. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of fractional order systems. In this paper, based on the residue harmonic balance method, a detailed analysis on approximations to the periodic oscillations of the fractional order Van der Pol equation is investigated. The relations that express the frequency and amplitude of the generated oscillations as functions of the orders and parameters are shown. Moreover, some examples are provided for comparing approximations with numerical solutions of the periodic oscillations. Numerical results reveal that the residue harmonic balance method is very effective for obtaining approximate solutions of fractional oscillations.     

A new state observer for two coupled Van Der Pol Oscillators

In this paper, first, a stable state observer is introduced for an isolated Van der Pol stable. This observer is extended for two unidirectional and bidirectional coupled Van der Pol Oscillators, and their convergence proved independently. The arrangement of observed states and observer gain is novel that guaranties the stability. Finally, the performances of the observers are investigated by numerical simulations.     

Model containing coupled subsystems with oscillations of different types

Consideration was given to an autonomous model containing coupled subsystems (MCCS) and, in the absence of couplings between the subsystems, falling down into systems of ordinary differential equations. It is assumed that the subsystems admit different types of nondegenerate single-frequency oscillations. Solved were the problems of oscillations in MCCS, their stability, and stabilization of MCCS oscillation by smooth autonomous coupling controls. It was shown how the developed theory is applied to the coupled Duffing and Van der Pol oscillators.     

Leader-following consensus for uncertain second-order nonlinear multi-agent systems

This paper studies the leader-following consensus problem for a class of second-order nonlinear multi-agent systems subject to linearly parameterized uncertainty and disturbance. The problem is solved by integrating the adaptive control technique and the adaptive distributed observer method. The design procedure is illustrated by an example with a group of Van der Pol oscillators as the followers and a harmonic system as the leader.     

不确定Van der Pol混沌系统的模糊同步控制

本文研究了不确定Van der Pol混沌系统的同步问题,并进行了基于规则的模糊逻辑控制器(FLC)的控制。首先寻找主从Van der Pol混沌系统满足Lyapunov稳定性理论的条件,在此基础上建立模糊规则,设计模糊控制器,实现不确定混沌系统的同步。通过不确定VanderPol混沌系统的两组仿真结果,验证了模糊同步控制方法具有很好的鲁棒性。最后为了进一步验证该方法的有效性,本文在相同条件下,利用反馈控制的方法实现不确定主从VanderPol混沌系统的同步,然后再将此方法的仿真结果与本文的模糊同步控制方法的仿真结果在稳态误差及同步所需时间这两个方面进行对比分析。分析结果验证了本文同步方法的可行性及有效性。     

A reactive coordination scheme for a many-robot system

This paper presents a novel approach for coordinating a homogeneous system of mobile robots using implicit communication in the form of broadcasts. The broadcast-based coordination scheme was developed for the Army Ant swarm-a system of small, relatively inexpensive mobile robots that can accomplish complex tasks by cooperating as a team. The primary drawback, however, of the Army Ant system is that the absence of a central supervisor poses difficulty in the coordination and control of the agents. Our coordination scheme provides a global "group dynamic" that controls the actions of each robot using only local interactions. Coordination of the swarm is achieved with signals we call "heartbeats". Each agent broadcasts a unique heartbeat and responds to the collective behavior of all other heartbeats. We generate heartbeats with van der Pol oscillators. In this application, we use the known properties of coupled van der Pol oscillators to create predictable group behavior. Some of the properties and behaviors of coupled van der Pol oscillators are discussed in detail. We emphasize the use of this scheme to allow agents to simultaneously perform an action such as lifting, steering, or changing speed. The results of experiments performed on three actual heartbeat circuits are presented and the behavior of the realized system is compared to simulated results. We also demonstrate the application of the coordination scheme to global speed control.     

On the reachable set of inflated singularly perturbed differential equations

Singularly perturbed differential equations with slow and fast subsystems are under consideration. It is well-known that the invariant probability measures of the unperturbed fast subsystems produce a limiting system for the slow subsystem. Unfortunately, the limiting system is non-smooth in general and might be too big since it also generates trajectories that are not related to the singularly perturbed system. However, if the flows produced by the unperturbed fast subsystems are chain transitive an inflation of the singularly perturbed system can be used in order to approximate all trajectories of the limiting system. Moreover, it turns out that the reachable sets of the limiting system are contained in the reachable sets of the slightly inflated singularly perturbed system.