本文選題：同步化　切入點(diǎn)：協(xié)同態(tài)　出處：《華東師范大學(xué)》2017年博士論文

【摘要】：同步化現(xiàn)象是指大量組元構(gòu)成的系統(tǒng)中,由于組元之間的相互協(xié)作或競爭,從而在宏觀上形成步調(diào)一致的現(xiàn)象,這種現(xiàn)象廣泛存在于自然界和人類社會中。理論上,研究同步化的經(jīng)典模型是Kuramoto模型。過去四十多年對Kuramoto類相振子模型的研究表明:在絕大多數(shù)情況下,系統(tǒng)的同步化,即從無序態(tài)向有序態(tài)(協(xié)同態(tài))的相變是二級相變。2011年,西班牙小組發(fā)現(xiàn)在一類廣義Kuramoto模型中,當(dāng)網(wǎng)絡(luò)的拓?fù)浣Y(jié)構(gòu)為無標(biāo)度網(wǎng),并且網(wǎng)絡(luò)的節(jié)點(diǎn)度與振子的本征頻率正相關(guān)時,系統(tǒng)的同步化相變是一個突然的、爆炸式的躍變,并且伴隨著磁滯現(xiàn)象,這種現(xiàn)象稱為爆炸式同步化現(xiàn)象。隨后對爆炸式同步化的研究迅速成為了復(fù)雜系統(tǒng)研究前沿的一個熱點(diǎn)。本文通過數(shù)值模擬和理論分析,對頻率權(quán)重耦合Kuramoto模型和頻率權(quán)重耦合Stuart-Landau振子模型進(jìn)行了系統(tǒng)的研究,主要結(jié)果包括:1、研究了雙峰洛倫茲頻率分布下的頻率權(quán)重耦合Kuramoto模型從連續(xù)性相變到爆炸式同步相變的轉(zhuǎn)換,解析給出了其向前相變的臨界點(diǎn);2、研究了均勻頻率分布下頻率權(quán)重耦合Kuramoto模型隨著耦合強(qiáng)度增加的相變行為,解析給出了其向前相變的臨界點(diǎn);3、首次發(fā)現(xiàn)了一種分離的、多集團(tuán)化的、時變的高階協(xié)同態(tài)——柏勒洛豐態(tài)(B態(tài))。在這種態(tài)中,系統(tǒng)形成多個分離的協(xié)同集團(tuán),每個集團(tuán)內(nèi)部振子的相位和瞬時頻率不是鎖定的,但平均頻率是相同的,它們形成有規(guī)律的階梯結(jié)構(gòu);4、在非對稱頻率分布下的頻率權(quán)重耦合Kuramoto模型和耦合擁護(hù)者和反對者的Kuramoto模型中觀察到了 B態(tài);5、研究了頻率權(quán)重耦合Stuart-Landau振子中的爆炸式振蕩死亡。發(fā)現(xiàn)了三種不同類型的爆炸式振蕩死亡現(xiàn)象;6、解析給出了頻率權(quán)重耦合Stuart-Landau振子系統(tǒng)向后相變點(diǎn),證明了其不隨本征頻率分布的變化而變化。以上結(jié)果表明耦合振子系統(tǒng)中可能廣泛存在與B態(tài)類似的高階協(xié)同態(tài),并且在具有振幅的耦合振子系統(tǒng)中同樣可以發(fā)生爆炸式同步現(xiàn)象。這些結(jié)果不僅加深了我們對耦合振子系統(tǒng)的協(xié)同行為的認(rèn)識,而且為研究實際網(wǎng)絡(luò)系統(tǒng)中的爆炸式同步化現(xiàn)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.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O19

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