耦合Stuart-Landau振子系统中的同步相变

发布时间：2018-06-30 06:02

本文选题：相变 + 同步化　；参考：《华东师范大学》2017年博士论文

【摘要】：很多自然界的现象都可以使用大量个体之间通过相互作用形成的集体行为来描述。现如今,随着大数据时代的来临,人们获得了更多关于这些个体以及其相互作用的信息。自然界中的个体往往被限制在一个有限范围内做连续运动,对于二维系统来说,这意味着这些个体必须具有不动点或周期性的行为特征。这些周期性行为称为极限环,这些个体则称为振子。由于个体内部往往存在一些非常复杂的非线性行为,因此需要使用比简谐振子更高级的模型来描述。极限环出现的一种重要机制就是霍普夫分岔,而Stuart-Landau振子是霍普夫分岔的标准形式。所以,研究大量Stuart-Landau振子的集体行为十分必要。本文中,我们主要研究了大量Stuart-Landau振子的同步相变问题。当一群振子通过相互作用耦合在一起,整个系统会随着耦合强度的逐渐增加从完全无序的状态达到所有振子完全同步振动的状态。以往的研究认为这种转变的过程是连续的,然而2011年西班牙的一个研究小组使用相对复杂的手段在相振子模型中展示了这种从无序到同步的变化可以是不连续的,称为爆炸式同步。我们发现通过调节Stuart-Landau振子之间反应耦合与耗散耦合的比例,可以实现从连续相变到不连续相变的转化。我们通过数值模拟、理论解析的方式分析了这种转变发生的原因。在反应耦合相对较强时,系统会存在一个完全无序态与同步态共存的参数范围。在无序态边缘增加耦合强度将导致其失去稳定而快速变化到同步态;在同步态边缘完全同步态并不存在,减少耦合强度将导致振子从霍普夫分岔变化到鞍结点分岔,使得系统同步程度发生连锁反应快速掉回到无序态。我们还发现了另外一种不连续相变行为,称之为老化猝死。当Stuart-Landau振子的霍普夫分岔参数为负时,称其处于死亡态。固定耦合强度,逐渐增加系统中处于死亡态振子的比例时,系统会从同步态转变到无序态,代表着系统从正常工作状态转变到失效状态,这种现象被称为老化。在较强的反应耦合下系统的老化行为是不连续的,系统会从能正常工作的状态突然变化到失效状态。初步研究表明这种老化猝死行为和爆炸式同步具有类似的机制。人类的耳蜗是将外部声波的机械信号转换为神经元的脉冲电信号的重要结构,其被证实具有非常灵敏的频率识别能力和微弱信号探测能力。研究清楚耳蜗是如何具有这些能力的不仅在仿生学上具有十分重要的意义,而且对治疗听力衰退具有指导作用。以往的研究大多着重于耳蜗的生理学结构,而对其底层物理机制缺乏了解。耳蜗内部的毛细胞振动行为可以使用Stuart-Landau振子来描述。我们构建了一个层级耦合模型来模拟耳蜗内部毛细胞在外界声波驱动下的振动行为,发现团簇状结构的毛细胞集团可以呈现不连续相变行为。这种不连续相变可以增强耳蜗探测特定声波频率的精准性,以及提高对微弱信号的响应。同时,我们说明了包含更多毛细胞的簇团具有更好的效果,以此来解释不同物种对声音信号的敏感程度。
[Abstract]:A lot of natural phenomena can be described by a large number of individuals through interaction between interactions. Now, with the advent of the big data age, more information about these individuals and their interaction is obtained. In a two-dimensional system, this means that these individuals have to have fixed point or periodic behavior characteristics. These periodic behavior is called the limit ring, and these individuals are called vibrators. Because there are often very complex nonlinear behaviors in the individual, a model that is more advanced than the simple harmonic oscillator is needed. One of the important mechanisms is the Hopf bifurcation, and the Stuart-Landau oscillator is the standard form of the Hopf bifurcation. Therefore, it is necessary to study the collective behavior of a large number of Stuart-Landau oscillators. In this paper, we mainly study the problem of synchronous phase transition of a large number of Stuart-Landau oscillators. In the previous study, the process of the transition is continuous, but in 2011, a team in Spain used relatively complex means to show this from disorder to synchronization in the phase oscillator model. The change can be discontinuous, called explosive synchronization. We find that by adjusting the ratio of the coupling and dissipative coupling between the Stuart-Landau oscillator, we can realize the transformation from the continuous phase to the discontinuous phase transition. We analyze the cause of this transformation by numerical simulation and theoretical analysis. When the system is strong, the system will have a parameter range of completely disordered state and synchronous state. The increase of coupling strength at the edge of disordered state will result in the loss of stability and rapid change to the synchronous state; the complete synchronous state in the edge of the synchronous state does not exist. The reduction of the coupling strength will lead to the change of the oscillator from the Hopf bifurcation to the saddle node bifurcation. We also found another kind of discontinuous phase transition, called aging sudden death. When the Stuart-Landau oscillator's Hopf bifurcation parameter is negative, it is known to be in the dead state. When the coupling strength is fixed and gradually increases the proportion of the dead vibrator in the system, the system will be in synchronization. The change of state to disorder represents the transition from normal working state to failure state. This phenomenon is called aging. Under strong reaction coupling, the aging behavior of the system is discontinuous, and the system will suddenly change from normal working state to failure state. There is a similar mechanism. The human cochlea is an important structure that converts the mechanical signals of external sound waves into neurons, and is proved to have very sensitive frequency identification and weak signal detection ability. It is very important not only to study how the cochlea is able to have these abilities, but also in bionics. Most of the previous studies have focused on the physiological structure of the cochlea, but lack of understanding of the underlying physical mechanism. The vibration behavior of the hair cells in the cochlea can be described by the Stuart-Landau oscillator. A hierarchical model is constructed to simulate the external sound waves in the inner cochlea. This discontinuous phase transition can enhance the accuracy of the specific acoustic frequency of the cochlea detection and improve the response to the weak signal. At the same time, we show that the clusters containing more hair cells have a better effect. The sensitivity of different species to sound signals.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O414.2

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