频率权重耦合振子模型中的协同行为

发布时间：2018-03-28 04:20

本文选题：同步化　切入点：协同态　出处：《华东师范大学》2017年博士论文

【摘要】：同步化现象是指大量组元构成的系统中,由于组元之间的相互协作或竞争,从而在宏观上形成步调一致的现象,这种现象广泛存在于自然界和人类社会中。理论上,研究同步化的经典模型是Kuramoto模型。过去四十多年对Kuramoto类相振子模型的研究表明:在绝大多数情况下,系统的同步化,即从无序态向有序态(协同态)的相变是二级相变。2011年,西班牙小组发现在一类广义Kuramoto模型中,当网络的拓扑结构为无标度网,并且网络的节点度与振子的本征频率正相关时,系统的同步化相变是一个突然的、爆炸式的跃变,并且伴随着磁滞现象,这种现象称为爆炸式同步化现象。随后对爆炸式同步化的研究迅速成为了复杂系统研究前沿的一个热点。本文通过数值模拟和理论分析,对频率权重耦合Kuramoto模型和频率权重耦合Stuart-Landau振子模型进行了系统的研究,主要结果包括:1、研究了双峰洛伦兹频率分布下的频率权重耦合Kuramoto模型从连续性相变到爆炸式同步相变的转换,解析给出了其向前相变的临界点;2、研究了均匀频率分布下频率权重耦合Kuramoto模型随着耦合强度增加的相变行为,解析给出了其向前相变的临界点;3、首次发现了一种分离的、多集团化的、时变的高阶协同态——柏勒洛丰态(B态)。在这种态中,系统形成多个分离的协同集团,每个集团内部振子的相位和瞬时频率不是锁定的,但平均频率是相同的,它们形成有规律的阶梯结构;4、在非对称频率分布下的频率权重耦合Kuramoto模型和耦合拥护者和反对者的Kuramoto模型中观察到了 B态;5、研究了频率权重耦合Stuart-Landau振子中的爆炸式振荡死亡。发现了三种不同类型的爆炸式振荡死亡现象;6、解析给出了频率权重耦合Stuart-Landau振子系统向后相变点,证明了其不随本征频率分布的变化而变化。以上结果表明耦合振子系统中可能广泛存在与B态类似的高阶协同态,并且在具有振幅的耦合振子系统中同样可以发生爆炸式同步现象。这些结果不仅加深了我们对耦合振子系统的协同行为的认识,而且为研究实际网络系统中的爆炸式同步化现象提供了思路。
[Abstract]:The phenomenon of synchronization refers to a large number of components in the system, because of the mutual cooperation or competition among the components, thus forming a step in the macro phenomenon, this phenomenon widely exists in the natural world and human society. The classical model for studying synchronization is the Kuramoto model. In the past 40 years, the study of Kuramoto phase oscillator model shows that in most cases, the phase transition from disordered state to ordered state (synergistic state) is a second-order phase transition. The Spanish team found that in a class of generalized Kuramoto models, when the topological structure of the network is scale-free and the node degree of the network is positively correlated with the intrinsic frequency of the oscillator, the synchronization phase transition of the system is a sudden and explosive jump. And with the phenomenon of hysteresis, this phenomenon is called explosive synchronization, and then the research of explosive synchronization becomes a hot spot in the research front of complex systems. The frequency weight coupled Kuramoto model and the frequency weight coupled Stuart-Landau oscillator model are studied systematically. The main results are as follows: 1. The transformation from continuous phase transition to explosive synchronous phase transition is studied in the frequency weighted coupled Kuramoto model with double peaks Lorentz frequency distribution. The critical point of forward phase transition is given analytically, and the phase transition behavior of frequency weighted coupled Kuramoto model with increasing coupling strength under uniform frequency distribution is studied. The critical point of forward phase transition is given analytically, and a kind of separation is found for the first time. A multigroup, time-varying, higher-order synergistic state, the Bellerofeng state, in which the system forms multiple separate synergistic clusters in which the phase and instantaneous frequency of the oscillator in each group are not locked, but the average frequency is the same. They form a regular ladder structure, and the B state 5 is observed in the frequency weighted coupled Kuramoto model under asymmetric frequency distribution and the Kuramoto model of coupling proponents and opponents. The explosion in the frequency weighted coupled Stuart-Landau oscillator is studied. Three different types of explosive oscillation death are found. The backward phase transition point of frequency weighted coupled Stuart-Landau oscillator system is given analytically. It is proved that it does not vary with the intrinsic frequency distribution. The above results indicate that there may be high order synergetic states similar to B states in the coupled oscillator system. Furthermore, explosive synchronization can also occur in coupled oscillator systems with amplitudes. These results not only deepen our understanding of the cooperative behavior of coupled oscillator systems, It also provides a train of thought for studying the explosive synchronization in the real network system.
【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】：O19

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