二维振荡系统中的斑图动力学研究

发布时间：2018-04-12 19:15

本文选题：振荡系统 + 复Ginzburg-Landau方程　；参考：《深圳大学》2017年硕士论文

【摘要】：在远离热力学平衡条件下可以产生非平衡斑图,非平衡斑图是在空间或时间上具有一定规律性的非均匀宏观结构,广泛存在于自然界。非平衡斑图可以是静止的,如图灵斑图,源于图灵对动物皮肤图样的思考;也可以是移动的,如行波、螺旋波、靶波。二维时空振荡系统中常见的两类周期性斑图有螺旋波和靶波,它们虽然在形貌上非常相似,都具有环状的周期性结构,但是各自的形成条件和波源是不相同的。螺旋波是普遍存在于自然界中的一种时空斑图,可以在均匀的时空系统中自发产生,它的波源是一个时空拓扑缺陷点;而靶波的形成与稳定存在需要局部不均匀区域持续性的激发,它的波源通常是一个区域。本文主要内容是通过数值模拟和理论分析相结合的方法来研究二维时空振荡系统中斑图动力学行为。第一章为前言,首先介绍了斑图动力学、反应扩散系统以及复GinzburgLandau方程(CGLE)的基本概念和背景知识,然后着重介绍了螺旋波与靶波斑图形成与传播的动力学行为,最后对耦合系统中的斑图动力学的研究进展与基本概念进行简单的介绍。第二章主要分析和讨论了CGLE系统中的能量特征值,该章节的主要内容有:(1)在CGLE系统中,系统的能量特征值可以分成两部分,分别对应的是螺旋波的能量特征值和一致振荡的能量特征值。(2)波的能量特征值为正表示波由波源向外传播,波的能量特征值为负则表示波由外向波源传播。(3)当能量特征值较大的波与能量特征值较小的波竞争时,能量特征值较大的波能够入侵能量特征值较小的波。新的波型能够产生并且稳定存在,其能量特征值必须大于原有的波型或者一致振荡能量特征值。第三章研究了二维非均匀振荡介质中波的竞争规律。在非均匀的介质中,当两边介质的色散关系曲线的斜率同号时,波的竞争规律按照均匀介质中的竞争模式进行。当两边介质的色散关系曲线的斜率为一正一负,且两条色散曲线相交于一点时,两边介质相连接的边界处产生界面选择波(Interface-Selected Wave-ISW)。ISW的频率高于一方为外传螺旋波的频率而低于一方为内传螺旋波的频率。当两边介质的色散关系曲线为一正一负,且不相交时,两边介质中的波共存。第四章主要研究了双层耦合系统中周期性振荡行为。本章节主要以双层耦合CGLE系统为时空模型,研究了不同频率的靶波耦合内传螺旋波产生的动力学行为,发现在弱耦合强度的条件下,响应系统的模会出现靶波,我们称之为振幅靶波。振幅靶波具有周期性振荡频率,我们利用快速傅里叶变换(FFT)着重分析了这种周期性振荡频率的来源,发现当内传的靶波耦合内传的螺旋波时,响应系统会出现新的频率,而振幅靶波的频率值正好是这个新的频率与原有驱动系统中靶波频率之差。此外,当外传的靶波耦合内传的螺旋波时,响应系统也会出现新的频率,而振幅靶波的频率值恰好是新的频率与原有驱动系统中靶波频率之和。最后,我们研究了不同频率的靶波耦合内传螺旋波时同步函数的变化规律。第五章是对本文研究工作的总结与展望。
[Abstract]:Under the conditions far from thermodynamic equilibrium can produce non-equilibrium pattern, nonequilibrium pattern has certain regularity in space or time on non uniform macro structure, widely exists in the nature. Non equilibrium pattern can be static, such as Turing pattern, derived from animal skin on the Turing pattern also thinking; can be mobile, such as traveling wave, spiral wave, target wave. Two dimensional space-time oscillation system of two kinds of periodic patterns are common spiral and target wave, although they are very similar in morphology, the periodic structure is circular, but their formation conditions and source of spiral wave is not the same. Is a kind of spatiotemporal pattern generally exists in the nature, can arise spontaneously in the space-time system uniform, its source is a spatio-temporal topological defects; and the formation and stability of target wave in inhomogeneous regions of persistent excitation, it The source is usually an area. The main content of this paper is combined with the method of numerical simulation and theoretical analysis to study the dynamic behavior of two-dimensional spatiotemporal pattern oscillation system. The first chapter is the introduction, first introduced the pattern dynamics, reaction diffusion system and complex GinzburgLandau equation (CGLE) of the basic concepts and background knowledge, then focuses on the spiral and target patterns form the dynamic behavior and communication, finally, research progress of pattern dynamics in a coupled system and basic concepts are briefly introduced. The second chapter mainly analyzes and discusses the characteristics of energy in CGLE system, the main contents of the chapters are: (1) in CGLE in the system, the energy eigenvalues of the system can be divided into two parts, corresponding to the energy characteristics of spiral wave value and consistent oscillation energy eigenvalue. (2) the wave energy characteristic value is expressed by wave The source of energy the characteristic wave spread outward, negative values from outside sources said wave spread. (3) when the wave energy eigenvalues and wave energy competition characteristics of large value smaller, energy eigenvalues larger wave energy eigenvalue to invade small wave. The wave of new type can produce and stable the energy eigenvalues, must be consistent or oscillation energy characteristic value is larger than the original wave type. The third chapter studies the competition law of two-dimensional inhomogeneous medium. The oscillation in inhomogeneous medium, when the slope of the dispersion curve of the media on both sides of the same sign, in accordance with the rules of competition wave medium competition mode the slope of the dispersion curve. When the medium is both positive and negative, and two dispersion curves intersect at one point, located on both sides of the border connected to the interface of choice of medium wave (Interface-Selected Wave-ISW).ISW frequency A party is higher than outside the spiral wave frequency and below the side for inward spiral wave frequency. When the dispersion curve of the media on both sides as a negative, and do not intersect, the coexistence of wave on both sides of the medium. The fourth chapter mainly studies the periodic oscillation behavior of double coupling system. This chapter mainly to double the coupled CGLE system for space time model, we investigate the dynamic behaviors of target wave coupled inward spiral wave of different frequency is generated, found in weak coupling condition, the system response mode will appear target wave, we call the target wave amplitude. The amplitude of target wave has a periodic oscillation frequency, we use fast Fourier transform (FFT) focuses on the analysis of the origin of the periodic oscillation frequency, found that the spiral wave wave coupling when the target spread spread when the response of the system will have a new frequency, and the amplitude of target wave frequency value is just the new frequency and The difference of the original target wave frequency drive system. In addition, when the spiral wave coupled wave spread out of the target, the response system will also appear in the new frequency and amplitude of target wave frequency value is just the wave frequency and the frequency of the new target and the original drive system. Finally, we studied the change rule of synchronization the target wave spread function coupling spiral wave with the same frequency. The fifth chapter is the summary and Prospect of this research work.

【学位授予单位】：深圳大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：N93

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