偏心摆轮的非线性研究.rar

摘  要：在原有波尔共振仪的基础上，设计了偏心摆轮振动模型,建立了该模型的运动学方程, 利用计算机进行了数值计算，分别对该模型进行了定性和定量研究。

　　　首先，通过计算做出了的分岔图，由分岔图可见，当时,系统都处于1周期; 时,为2周期运动状态;混沌区域频率大致为。同时也做了时间序列图，所得的结论与分岔图基本一致。另外通过对系统初始条件的微小改变，研究了系统处于混沌运动状态下，对初始条件的敏感性。可以看出在两种情况下系统经历了短暂的重合之后，迅速反应出了极大的差别。

　　　对应于不同的选取了100种情况，分别研究了近似熵和关联维。在近似熵的研究中可以发现，当时熵值较小，此时系统处于简单的周期运动，但时，近似熵值较大，这说明系统的运动状态较复杂，但有所波动回落，这反映在分岔图上有周期性窗口。时，近似熵值又回落，这从分岔图上可以看出系统又归于稳定的周期运动。同为周期运动，它的近似熵值比时要大，这反映了随着的变化，系统的整体的复杂性有所提高。

　　　在关联维的研究中可以发现，时关联维值较小，此时系统处于简单的周期运动，关联维值在1.0左右，但时，关联维值较大，这说明系统的运动状态较复杂，在混沌区域关联维数达到最大，但也存在波动回落，这反映在分岔图上有周期性窗口。时，关联维值又回落，关联维值在1.0左右，可见又处于简单的周期运动。

　　　总之，经波尔共振仪改装的偏心摆轮装置具有周期混沌运动特性，无论从定性还是定量均得到证明。

关键词:偏心摆轮；分岔；时间序列；近似熵；关联维；混沌

Abstract：On the basis of the Boer resonance instrument, an oscillation device of eccentric balance wheel is designed .The Kinematic equations to the model is constructed and the numerical calculation is done with comtuers. The study of qualitative and quantitative is done to the model.

At first, the bifurcation diagram of  is done. From the diagram ,we can see that the system is in a cycle while the  ,and in two cycle while .When the ,the system is in chaos. At the same time ,the diagram of the time series is done and the result is consistent with  the bifuracation. In addition , the initial conditions is slight changed , we study the sensitivty of the system to this.It can be observed that the system in two slight differt initial conditons show great different afer temparary coincidence.

　　The ApEn and CD are stdudied to 100 different values of . In the study of the ApEn, We can see that, the ApEn is little when the .It demostrate that the system is now in simple period state. But when ,the Apen is larger. It demostrate that the system is in complex state. During the peiod , the ApEn fluctuate down , this can be explain by the periodic window in the bifurcation diagram. When the ,the ApEn become down again, this demonstrate that the system goes back to the periodic state again. The ApEn of the later is larger than the former,this declares that the system becomes more complex as the   is increased.

　　In the study of the CD, We can see that, the value of CD is around 1.0, when  .But when ,the CD is larger. It demostrate that the system is in more complex state. When the , the system is in chaotic state,the value of CD is largest. During the chaotic state, the fluctuate down can be observed.This can be explain by the period windows in the in the bifurcation diagram. When the ,the values of CD go back to 1.0 again. To this ,we can see that the system is in period state again.

　　In a word, an oscillation device of eccentric balance wheel desigened from the Boer resonance instrument has character of  period and chaotic .This can be demonstrated by the qualitative side and quantitative side.

Keywords：eccentric balance wheel; bifurcation;time series;approximate entropy;correlation dimensions; chaos