基于正弦映射的混沌理论及其应用研究
发布时间：2018-12-24 12:03
【摘要】：随着研究工作的进行,很多种不同的簇发形式被提出,但是大部分簇发现象都只含有一个或者两个振荡分支,很少见多振荡分支的簇发现象;而且像这样的簇发现象大部分都是在高阶连续系统中发现的,有学者就提出了采用离散系统去模拟这种简单的簇发现象,主要是由于离散系统在数值仿真的时候比连续系统更方便,可以在短时间内得到大量的数据。但是现有的离散模型也不能够模拟更为复杂的多分支簇发振荡现象,所以有必要进一步研究关于离散系统中的多分支簇发振荡现象及其产生机理。本文以一个正弦映射为研究对象,通过研究一维正弦系统的动力学特征研究发现其具有初值敏感性、倍周期分岔和对称性破缺分岔等非线性现象,但是这些性质不足以产生多分支簇发振荡现象。为了采用离散系统模拟多分支簇发振荡现象,我们进一步采用正弦映射和一个三次方映射,通过非线性耦合方式组合成一个二维的离散系统,通过分析发现此系统中存在多种不同的分岔特性,但是此系统为单一尺度系统,也不能够产生多分支簇发振荡现象。根据以上分析本文最后构造了三维离散系统,通过对其平衡点的稳定性分析,得到了其快子系统发生Fold分岔和Neimarker-sacker分岔的参数条件。通过数值计算得到了系统在三种不同参数条件下,呈现三种不同的快慢簇发振荡现象,发现此簇发振荡由多个快慢振荡的分支所组成,且每个分支由其独特的簇发振荡现象,进一步采用Rinzel快慢分析法给出了三种不同的多分支快慢簇发振荡现象及其相应分支的产生机理。根据三种不同快慢振荡现象的产生机理,文中对这三种簇发振荡现象进行了分类。最后设计了一组相应的电路实验得到了三种不同快慢振荡现象的相图,从而验证了理论分析与数值计算的正确性。
[Abstract]:With the development of the research work, many different clusters have been proposed, but most of the cluster phenomena only contain one or two oscillatory branches, and it is rare to find clusters with multiple oscillatory branches. Moreover, most of the cluster discovery images like this are found in high-order continuous systems. Some scholars have proposed to use discrete systems to simulate this simple cluster discovery image. The main reason is that the discrete system is more convenient than the continuous system in numerical simulation, and a large amount of data can be obtained in a short time. However, the existing discrete models can not simulate the more complex multi-branching cluster oscillation phenomenon, so it is necessary to further study the multi-branch cluster oscillation phenomenon and its mechanism in discrete systems. In this paper, a sinusoidal mapping is studied. By studying the dynamic characteristics of one dimensional sinusoidal system, it is found that it has some nonlinear phenomena, such as initial value sensitivity, period doubling bifurcation, symmetry breaking bifurcation and so on. However, these properties are not sufficient to produce multi-branch cluster oscillation. In order to simulate the multi-branching cluster oscillations with discrete systems, we further use sinusoidal maps and a cubic map to form a two-dimensional discrete system by nonlinear coupling. It is found that there are many different bifurcation characteristics in this system, but the system is a single scale system and can not produce multi-branch cluster oscillation. Based on the above analysis, a three dimensional discrete system is constructed. By analyzing the stability of its equilibrium point, the parameter conditions for Fold bifurcation and Neimarker-sacker bifurcation in its fast subsystem are obtained. By numerical calculation, three different fast and slow cluster oscillations are obtained under three different parameter conditions. It is found that the cluster oscillation is composed of several fast and slow oscillating branches, and each branch is composed of its unique cluster oscillation phenomenon. Furthermore, three different multi-branch fast and slow cluster oscillations and their corresponding branching mechanisms are given by using the Rinzel fast and slow analysis method. According to the generation mechanism of three different fast and slow oscillation phenomena, the three cluster oscillation phenomena are classified in this paper. Finally, a set of corresponding circuit experiments are designed to obtain three phase diagrams of different fast and slow oscillation phenomena, which verifies the correctness of theoretical analysis and numerical calculation.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O415.5;TB30
，
本文编号：2390595
[Abstract]:With the development of the research work, many different clusters have been proposed, but most of the cluster phenomena only contain one or two oscillatory branches, and it is rare to find clusters with multiple oscillatory branches. Moreover, most of the cluster discovery images like this are found in high-order continuous systems. Some scholars have proposed to use discrete systems to simulate this simple cluster discovery image. The main reason is that the discrete system is more convenient than the continuous system in numerical simulation, and a large amount of data can be obtained in a short time. However, the existing discrete models can not simulate the more complex multi-branching cluster oscillation phenomenon, so it is necessary to further study the multi-branch cluster oscillation phenomenon and its mechanism in discrete systems. In this paper, a sinusoidal mapping is studied. By studying the dynamic characteristics of one dimensional sinusoidal system, it is found that it has some nonlinear phenomena, such as initial value sensitivity, period doubling bifurcation, symmetry breaking bifurcation and so on. However, these properties are not sufficient to produce multi-branch cluster oscillation. In order to simulate the multi-branching cluster oscillations with discrete systems, we further use sinusoidal maps and a cubic map to form a two-dimensional discrete system by nonlinear coupling. It is found that there are many different bifurcation characteristics in this system, but the system is a single scale system and can not produce multi-branch cluster oscillation. Based on the above analysis, a three dimensional discrete system is constructed. By analyzing the stability of its equilibrium point, the parameter conditions for Fold bifurcation and Neimarker-sacker bifurcation in its fast subsystem are obtained. By numerical calculation, three different fast and slow cluster oscillations are obtained under three different parameter conditions. It is found that the cluster oscillation is composed of several fast and slow oscillating branches, and each branch is composed of its unique cluster oscillation phenomenon. Furthermore, three different multi-branch fast and slow cluster oscillations and their corresponding branching mechanisms are given by using the Rinzel fast and slow analysis method. According to the generation mechanism of three different fast and slow oscillation phenomena, the three cluster oscillation phenomena are classified in this paper. Finally, a set of corresponding circuit experiments are designed to obtain three phase diagrams of different fast and slow oscillation phenomena, which verifies the correctness of theoretical analysis and numerical calculation.
【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O415.5;TB30
，
本文编号：2390595
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