CHAOS IN THE SOFTENING DUFFING SYSTEM UNDER MULTI-FREQUENCY PERIODIC FORCES

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincaré map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The (Melnikov's) global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.

Demodulation of acoustic telemetry binary phase shift keying signal based on high-order Duffing system

In order to grasp the downhole situation immediately, logging while drilling(LWD) technology is adopted. One of the LWD technologies, called acoustic telemetry, can be successfully applied to modern drilling. It is critical for acoustic telemetry technology that the signal is successfully transmitted to the ground. In this paper, binary phase shift keying(BPSK) is used to modulate carrier waves for the transmission and a new BPSK demodulation scheme based on Duffing chaos is investigated. Firstly, a high-order system is given in order to enhance the signal detection capability and it is realized through building a virtual circuit using an electronic workbench(EWB). Secondly, a new BPSK demodulation scheme is proposed based on the intermittent chaos phenomena of the new Duffing system. Finally, a system variable crossing zero-point equidistance method is proposed to obtain the phase difference between the system and the BPSK signal. Then it is determined that the digital signal transmitted from the bottom of the well is ‘0’ or ‘1’. The simulation results show that the demodulation method is feasible.

BIFURCATION IN A NONLINEAR DUFFING SYSTEM WITHMULTI-FREQUENCY EXTERNAL PERIODIC FORCES

By introducing nonlinear frequency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.

Stochastic period-doubling bifurcation in biharmonic driven Duffing system with random parameter

Stochastic period-doubling bifurcation is explored in a forced Duffing system with a bounded random parameter as an additional weak harmonic perturbation added to the system. Firstly, the biharmonic driven Duffing system with a random parameter is reduced to its equivalent deterministic one, and then the responses of the stochastic system can be obtained by available effective numerical methods. Finally, numerical simulations show that the phase of the additional weak harmonic perturbation has great influence on the stochastic period-doubling bifurcation in the biharmonic driven Duffing system. It is emphasized that, different from the deterministic biharmonic driven Duffing system, the intensity of random parameter in the Duffing system can also be taken as a bifurcation parameter, which can lead to the stochastic period-doubling bifurcations.

基于Duffing振子的铁路轨道移频信号测量系统设计

铁路轨道作为移频信号的传输载体,其对于信号对象传输轨迹疏密度的辨别能力,决定了轨道体系对移频信号的测量准确性;为实现对信号对象传输频率的准确辨别,设计基于Duffing振子的铁路轨道移频信号测量系统;将电源模块输出的电量信号,按需分配至STM32F103微处理器、移频信号辨别模块与DSP测量单元之中,完善各级应用部件之间的连接关系,实现铁路轨道移频信号测量系统的硬件方案设计;确定移频信号的Duffing振子描述条件,针对其定义形式,构造移频信号的相空间,完成对铁路轨道移频信号的提取;求解铁路轨道的Lyapunov指数,并构建移频信号的Duffing方程,确定混沌测量参数的取值范围,实现对铁路轨道移频信号的测量,联合各级应用部件,完成基于Duffing振子的铁路轨道移频信号测量系统设计;实验结果表明,上述系统测量所得信号传输轨迹疏密度与真实轨迹疏密度相似度较高,并且测量结果准确率最高值达到了84%左右,说明其能够实现对铁路轨道移频信号的准确测量。

准周期激励Duffing系统中奇异非混沌吸引子的判别方法

奇异非混沌动力学是非线性动力学领域中的新课题。本文以准周期激励Duffing振子为例,对其产生的奇异非混沌吸引子(strange nonchaotic attractors,SNAs)进行分析。通过三维庞加莱截面和定量方法如傅里叶变换、李雅普诺夫指数、李雅普诺夫维数、关联维数和盒维数检测SNAs是否存在。研究结果表明,傅里叶变换无法判断混沌与奇异非混沌行为。而李雅普诺夫指数、李雅普诺夫维数可以作为检测系统混沌与非混沌指标。关联维数和盒维数显著表明系统奇异与非奇异性,从而阐明适用于准周期驱动Duffing振子中存在SNAs的判别方法,并为其他类似系统检测SNAs提供指导。

Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales

The dynamical behavior on fractional-order Duffing system with two time scales is investigated,and the point-cycle coupling type cluster oscillation is firstly observed herein.When taking the fractional order as bifurcation parameter,the dynamics of the autonomous Duffing system will become more complex than the corresponding integer-order one,and some typical phenomenon exist only in the fractional-order one.Different attractors exist in various parameter space,and Hopf bifurcation only happens while fractional order is bigger than 1 under certain parameter condition.Moreover,the bifurcation behavior of the autonomous system may regulate dynamical phenomenon of the periodic excited system.It results into the point-cycle coupling type cluster oscillation when the fractional order is bigger than 1.The related generation mechanism based on slow-fast analysis method is that the slow variation of periodic excitation makes the system periodically visit different attractors and critical points of different bifurcations of the autonomous system.

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology, the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research. In this paper, the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively inves- tigated. With the stability criterion of linear fractional systems, the synchronization of a fractional non-autonomous system is obtained. Specifically, an effective singly active control is proposed and used to synchronize a fractional order Duffing system. The nu- merical results demonstrate the effectiveness of the proposed methods.

Analysis of a kind of Duffing oscillator system used to detect weak signals

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

基于CFAR和Duffing振子的FOD联合检测算法

机场跑道异物(Foreign Object Debris,FOD)对飞行器的安全运行危害巨大。图像处理识别目标,容易受到天气、光照的影响;毫米波雷达中常用的杂波图CFAR检测算法由于机场跑道的散射,在信杂比较低时,存在虚警较大的问题。针对现有的问题,提出一种CFAR与Duffing振子相结合的双层异物检测算法。利用传统的杂波图CFAR抑制杂波、设置阈值处理后,得到含有虚警信号的FOD回波信号,并提取FOD目标信号或者是虚警信号对应的频率信息,生成Duffing振子对应的本帧信号;再对FOD回波信号进行检测以剔除虚警信号,筛选出目标信号。通过Monte Carlo仿真验证了算法的有效性。

乘性色噪声激励下广义分数阶van der Pol-Duffing振子的随机分岔分析

本文研究了在乘性色噪声激励下含分数阶导数项的广义Duffing振子的随机分岔.首先,利用一种回复力和阻尼力的线性组合等效替换系统中的分数阶导数项;其次,对系统中的三次项进行线性化处理,利用最小均方误差原理,将系统转变成整数阶系统,由随机平均法求得系统的稳态概率密度函数;最后,通过拟不可积Hamilton系统随机平均法得到系统不变测度的最大Lyapunov指数,并对系统进行随机D-分岔和P-分岔分析.研究发现,分数阶导数阶数、噪声的自相关时间等参数的改变可以诱发系统发生随机P-分岔.

周期激励下广义离散Duffing系统的多稳态分析

针对一类周期激励下的广义离散Duffing系统,运用快慢分析方法,对系统状态进行数值模拟,通过分岔图和时间历程图对系统进行分析,得到不同参数下系统所表现出的新型簇发振荡模式,并探讨其与连续Duffing系统之间的联系.系统的簇发振荡模式被分为两类,一类是当慢变量穿过Fold分岔点或混沌激变点,吸引子发生转迁所诱发的各种对称式簇发振荡,另一类则是当慢变量无法穿过Fold分岔点或混沌激变点,由延迟Flip分岔所诱发的各种非对称式簇发振荡.

小阻尼时变双项位势Duffing型方程的调和解

本文利用弯曲扭转拓扑映射的不动点定理,证明了如下结果:若a(t),b(t)是连续的T-周期函数,则小阻尼Duffing型方程x″+εx′+a(t)f(x)+b(t)g(x)=0至少存在四个调和解。

Parameter estimation method for a linear frequency modulation signal with a Duffing oscillator based on frequency periodicity

In view of the complexity of existing linear frequency modulation(LFM)signal parameter estimation methods and the poor antinoise performance and estimation accuracy under a low signal-to-noise ratio(SNR),a parameter estimation method for LFM signals with a Duffing oscillator based on frequency periodicity is proposed in this paper.This method utilizes the characteristic that the output signal of the Duffing oscillator excited by the LFM signal changes periodically with frequency,and the modulation period of the LFM signal is estimated by autocorrelation processing of the output signal of the Duffing oscillator.On this basis,the corresponding relationship between the reference frequency of the frequencyaligned Duffing oscillator and the frequency range of the LFM signal is analyzed by the periodic power spectrum method,and the frequency information of the LFM signal is determined.Simulation results show that this method can achieve high-accuracy parameter estimation for LFM signals at an SNR of-25 dB.

Duffing方程的遥远概周期解

本文通过指数二分性和上下解方法,在比较自然的条件下,证明了二阶非线性常微分方程u″+cu′+g(u)=p(t)的遥远概周期解的存在性,其中g∈C1(R),而p是遥远概周期函数。

Duffing振子在水工结构动力响应信号检测应用中的一种混沌判据

【目的】Duffing振子是一种典型的非线性动力系统,混沌状态的判断是利用Duffing振子进行水工结构动力响应信号检测的基础准则。Duffing振子系统进入混沌状态时,系统状态时序图和相图会变得更为无序和复杂,可以对系统状态进行直观反映,但直接将二者作为系统混沌状态的判断依据,存在较为模糊和主观性较大的问题。为了对Duffing振子系统状态时序图和相图的复杂程度进行定量描述,以便更准确地利用二者对Duffing振子系统混沌状态的判断,【方法】提出了一种利用系统内状态量的时序图和相图盒维数的指数组合的指标,将该指标是否发生突变作为利用Duffing振子进行信号检测时的混沌状态的判断依据。【结果】所提出的Duffing振子混沌状态识别指标在Duffing方程的仿真分析、正弦波仿真信号检测和悬臂梁锤击动力响应信号检测试验中分别表现出22.611%、17.514%和30.975%的相对变化率。同时振子对应的李雅普诺夫指数也由负转正,但Duffing方程的仿真分析结果显示李雅普诺夫指数作为识别指标更为敏感,在振子系统的时序图和相图未明显表现出混沌状态时,其已经由负转正。【结论】由试验结果可知:可以根据所提出识别指标是否发生突变进行识别振子混沌状态,达到信号检测的目的。同时,所提出的混沌状态识别指标以时序图和相图的复杂程度直接作为判断标准,与其他指标相比更为直接和合理。悬臂梁动力响应信号检测试验初步验证了所选取的识别指标应用到具体工作中的可行性,有望应用于结构损伤识别,共振发现等工作中。

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.

EMD及Duffing振子在小电流系统故障选线方法中的应用

针对小电流系统发生故障时,各线路零序电流的非平稳、非线性等复杂特性,给出一种基于经验模态分解(empirical mode decomposition,EMD)和Duffing振子检测的方法。EMD方法具有很好的自适应性,而Duffing振子系统对于与内驱动力同频的外界信号具有高度敏感性。首先对故障零序电流进行EMD分解,提取出第1个内禀模态函数作为故障零序电流特征量;然后提出特征量中的五次谐波分量,将其输入到Duffing振子中,通过系统相图变化来确定当前线路是否发生故障。对各条线路进行不同条件下故障模拟,仿真结果证明了该算法的可靠性。

基于Duffing振子的弱正弦信号检测方法研究

数值仿真发现,通过设置Duffing振子初值和参考信号初相使其满足一定关系,Duffing振子从大周期态向混沌态的转变(正向相变),比较其从混沌态向大周期态的转变(逆向相变),对周期策动力的幅值体现出了更稳健的敏感性.据此,提出了一种利用Duffing振子逆向相变实现弱正弦信号检测的方法,分析了其可行性,并通过仿真与传统的检测方法进行了对比.数据表明,相同条件下,所提检测方法具有更高的检测精度.此外,根据相空间分布特点,提出了一种相轨迹运动状态的判别方法.该判别方法计算量小,并具有较好的实时性,通过仿真验证了该方法的正确性.

联合增益递推的Duffing系统弱信号检测算法

相变判别在Duffing振子弱信号检测中具有至关重要的作用,传统的判别算法复杂度高而精度低,限制了Duffing振子的应用,因而需要寻求更优的判别算法.本文通过欧拉方程建立正弦策动力Duffing振子的状态方程,并对Duffing系统进行滤波估计,推导出系统扩展卡尔曼增益的递推表达形式,以此为据提出了一种联合卡尔曼增益递推方程判别Duffing系统状态的新方法,同时根据增益随信号幅度变化的规律,提出了运用联合系统对弱信号进行定性和定量检测的方法,仿真验证表明,该方法可有效地对弱信号进行检测,相比传统Lyapunov指数判别法,运算耗时降低了两个数量级,且运算精度得到了显著提高.