本文選題：相變 + 同步化　； 參考：《華東師范大學(xué)》2017年博士論文

【摘要】：很多自然界的現(xiàn)象都可以使用大量個(gè)體之間通過相互作用形成的集體行為來描述�，F(xiàn)如今,隨著大數(shù)據(jù)時(shí)代的來臨,人們獲得了更多關(guān)于這些個(gè)體以及其相互作用的信息。自然界中的個(gè)體往往被限制在一個(gè)有限范圍內(nèi)做連續(xù)運(yùn)動(dòng),對(duì)于二維系統(tǒng)來說,這意味著這些個(gè)體必須具有不動(dòng)點(diǎn)或周期性的行為特征。這些周期性行為稱為極限環(huán),這些個(gè)體則稱為振子。由于個(gè)體內(nèi)部往往存在一些非常復(fù)雜的非線性行為,因此需要使用比簡(jiǎn)諧振子更高級(jí)的模型來描述。極限環(huán)出現(xiàn)的一種重要機(jī)制就是霍普夫分岔,而Stuart-Landau振子是霍普夫分岔的標(biāo)準(zhǔn)形式。所以,研究大量Stuart-Landau振子的集體行為十分必要。本文中,我們主要研究了大量Stuart-Landau振子的同步相變問題。當(dāng)一群振子通過相互作用耦合在一起,整個(gè)系統(tǒng)會(huì)隨著耦合強(qiáng)度的逐漸增加從完全無序的狀態(tài)達(dá)到所有振子完全同步振動(dòng)的狀態(tài)。以往的研究認(rèn)為這種轉(zhuǎn)變的過程是連續(xù)的,然而2011年西班牙的一個(gè)研究小組使用相對(duì)復(fù)雜的手段在相振子模型中展示了這種從無序到同步的變化可以是不連續(xù)的,稱為爆炸式同步。我們發(fā)現(xiàn)通過調(diào)節(jié)Stuart-Landau振子之間反應(yīng)耦合與耗散耦合的比例,可以實(shí)現(xiàn)從連續(xù)相變到不連續(xù)相變的轉(zhuǎn)化。我們通過數(shù)值模擬、理論解析的方式分析了這種轉(zhuǎn)變發(fā)生的原因。在反應(yīng)耦合相對(duì)較強(qiáng)時(shí),系統(tǒng)會(huì)存在一個(gè)完全無序態(tài)與同步態(tài)共存的參數(shù)范圍。在無序態(tài)邊緣增加耦合強(qiáng)度將導(dǎo)致其失去穩(wěn)定而快速變化到同步態(tài);在同步態(tài)邊緣完全同步態(tài)并不存在,減少耦合強(qiáng)度將導(dǎo)致振子從霍普夫分岔變化到鞍結(jié)點(diǎn)分岔,使得系統(tǒng)同步程度發(fā)生連鎖反應(yīng)快速掉回到無序態(tài)。我們還發(fā)現(xiàn)了另外一種不連續(xù)相變行為,稱之為老化猝死。當(dāng)Stuart-Landau振子的霍普夫分岔參數(shù)為負(fù)時(shí),稱其處于死亡態(tài)。固定耦合強(qiáng)度,逐漸增加系統(tǒng)中處于死亡態(tài)振子的比例時(shí),系統(tǒng)會(huì)從同步態(tài)轉(zhuǎn)變到無序態(tài),代表著系統(tǒng)從正常工作狀態(tài)轉(zhuǎn)變到失效狀態(tài),這種現(xiàn)象被稱為老化。在較強(qiáng)的反應(yīng)耦合下系統(tǒng)的老化行為是不連續(xù)的,系統(tǒng)會(huì)從能正常工作的狀態(tài)突然變化到失效狀態(tài)。初步研究表明這種老化猝死行為和爆炸式同步具有類似的機(jī)制。人類的耳蝸是將外部聲波的機(jī)械信號(hào)轉(zhuǎn)換為神經(jīng)元的脈沖電信號(hào)的重要結(jié)構(gòu),其被證實(shí)具有非常靈敏的頻率識(shí)別能力和微弱信號(hào)探測(cè)能力。研究清楚耳蝸是如何具有這些能力的不僅在仿生學(xué)上具有十分重要的意義,而且對(duì)治療聽力衰退具有指導(dǎo)作用。以往的研究大多著重于耳蝸的生理學(xué)結(jié)構(gòu),而對(duì)其底層物理機(jī)制缺乏了解。耳蝸內(nèi)部的毛細(xì)胞振動(dòng)行為可以使用Stuart-Landau振子來描述。我們構(gòu)建了一個(gè)層級(jí)耦合模型來模擬耳蝸內(nèi)部毛細(xì)胞在外界聲波驅(qū)動(dòng)下的振動(dòng)行為,發(fā)現(xiàn)團(tuán)簇狀結(jié)構(gòu)的毛細(xì)胞集團(tuán)可以呈現(xiàn)不連續(xù)相變行為。這種不連續(xù)相變可以增強(qiáng)耳蝸探測(cè)特定聲波頻率的精準(zhǔn)性,以及提高對(duì)微弱信號(hào)的響應(yīng)。同時(shí),我們說明了包含更多毛細(xì)胞的簇團(tuán)具有更好的效果,以此來解釋不同物種對(duì)聲音信號(hào)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.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O414.2

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