本文選題：平衡點(diǎn)穩(wěn)定性 + HR神經(jīng)元��； 參考：《蘭州交通大學(xué)》2017年碩士論文

【摘要】：近些年以來,神經(jīng)元網(wǎng)絡(luò)系統(tǒng)已經(jīng)廣泛應(yīng)用于生物科學(xué)、計(jì)算機(jī)科學(xué)、工程技術(shù)以及物理科學(xué)等多個(gè)領(lǐng)域,并且隨著科學(xué)計(jì)算水平、控制理論技術(shù)、傳感器測(cè)試水平的飛速提高。神經(jīng)元網(wǎng)絡(luò)動(dòng)力學(xué)問題的研究開始引起了越來越多的專家以及學(xué)者的廣泛關(guān)注。而有關(guān)高維非線性問題的研究難度較大,也更具挑戰(zhàn)性和實(shí)用性,同時(shí)研究者們發(fā)現(xiàn)在實(shí)際應(yīng)用中系統(tǒng)存在反映滯后這一現(xiàn)象,而且發(fā)現(xiàn)反映滯后現(xiàn)象對(duì)大多數(shù)非線性系統(tǒng)的平衡點(diǎn)穩(wěn)定性的影響較為敏感,使得系統(tǒng)出現(xiàn)分岔和混沌等復(fù)雜的動(dòng)力學(xué)行為,由此根據(jù)理論知識(shí)和實(shí)際需要把時(shí)滯引入所研究的非線性系統(tǒng)中,研究其對(duì)系統(tǒng)各種穩(wěn)定性的影響。研究者還發(fā)現(xiàn)時(shí)滯對(duì)神經(jīng)元網(wǎng)絡(luò)系統(tǒng)的非線性動(dòng)力學(xué)行為特征等的影響更為復(fù)雜,探索起來也更有難度,為此,學(xué)術(shù)界掀起了對(duì)時(shí)滯神經(jīng)網(wǎng)絡(luò)研究的熱潮。本篇文章分析了兩類時(shí)滯神經(jīng)元網(wǎng)絡(luò)系統(tǒng)的平衡點(diǎn)穩(wěn)定性問題,并推出了這兩個(gè)模型發(fā)生Hopf分岔的相關(guān)條件,還對(duì)其中的部分理論進(jìn)行了簡(jiǎn)單的數(shù)值模擬。其主要研究?jī)?nèi)容以及創(chuàng)新之處敘述如下:首先,主要是對(duì)本篇文章所研究的HR和Hopfield這兩類神經(jīng)元網(wǎng)絡(luò)系統(tǒng)的發(fā)展史、研究現(xiàn)狀以及研究意義進(jìn)行概述,從而使得讀者對(duì)兩類神經(jīng)網(wǎng)絡(luò)系統(tǒng)有更具深入的了解,為后續(xù)的研究工作提供方便。其次,文章簡(jiǎn)單地介紹了后續(xù)研究所需要的相關(guān)定理及定義。然后,主要根據(jù)Hindmash和Rose提出的HR神經(jīng)網(wǎng)絡(luò)模型和相關(guān)文獻(xiàn)的建模方法,為其加入新的時(shí)滯建立了一個(gè)新的單時(shí)滯神經(jīng)元網(wǎng)絡(luò)模型。根據(jù)根與系數(shù)的密切關(guān)系詳細(xì)的敘述了所建立的新模型的正平衡點(diǎn)存在條件,并且應(yīng)用線性化理論和Hassard方法借助于規(guī)范性理論及中心流形定理推出了該模型在正平衡點(diǎn)處發(fā)生Hopf分岔的條件及判定Hopf分岔的分岔周期、分岔方向的判定表達(dá)式。應(yīng)用數(shù)學(xué)軟件模擬出相應(yīng)的時(shí)間歷程圖和有代表性的相圖。最后,由于考慮到高維非線性理論更具有實(shí)用性,因此,選取了Hopfield這一特別地四維神經(jīng)元網(wǎng)絡(luò)進(jìn)行深入的研究。主要?jiǎng)?chuàng)新之處是根據(jù)已有模型和有關(guān)理論知道神經(jīng)元之間具有相互作用和影響,并且在作用過程中也都存在反映滯后現(xiàn)象,所以在原有模型的基礎(chǔ)上加入了兩個(gè)長(zhǎng)連接、一個(gè)互為反向連接和相應(yīng)的時(shí)滯又得到了一個(gè)新的系統(tǒng),也就是本文要研究的第二個(gè)模型。這里與第一個(gè)模型的研究方法有幾個(gè)不同之處。區(qū)別一,是由于系統(tǒng)比較特別直接就能計(jì)算出該模型必有一個(gè)平衡點(diǎn)為原點(diǎn),不需要再對(duì)非負(fù)平衡點(diǎn)進(jìn)行平移了;區(qū)別二,是由于系統(tǒng)有多個(gè)時(shí)滯研究起來比較困難根據(jù)有關(guān)理論做一個(gè)等價(jià)變換,把原來的系統(tǒng)模型變換成只含有一個(gè)時(shí)滯的簡(jiǎn)單模型。然后再應(yīng)用與上一個(gè)模型基本相同的處理方法、定理和定義,探討該模型零平衡點(diǎn)穩(wěn)定性及其Hopf分岔的存在性,并且推導(dǎo)出了Hopf分岔點(diǎn)的參數(shù)表達(dá)式,得出分支點(diǎn)的分岔的方向和運(yùn)動(dòng)軌道的周期等相關(guān)性質(zhì)的判別式,還運(yùn)用數(shù)學(xué)軟件對(duì)該模型的穩(wěn)定性理論進(jìn)行了數(shù)值檢驗(yàn),進(jìn)一步證明了該部分理論的合理性。
[Abstract]:In recent years, the neural network system has been widely used in many fields, such as biological science, computer science, engineering technology and physical science. With the scientific computing level, the control theory and the rapid improvement of the sensor testing level, the research of neural network dynamics has begun to cause more and more experts. The research on high dimensional nonlinear problems is more difficult and more challenging and practical. At the same time, the researchers found that the system has the phenomenon of lagging in the practical application, and it is found that the lag phenomenon is more sensitive to the stability of the equilibrium point of most nonlinear systems, making the system more sensitive. There are complex dynamic behaviors such as bifurcation and chaos, thus introducing time-delay into the nonlinear system studied in the light of theoretical knowledge and practical needs, and studying its influence on the stability of the system. The researchers also find that the effect of time delay on the nonlinear dynamic line of neural network system is more complex and is explored. In this paper, the stability of the equilibrium point of two kinds of neural network systems with time delay is analyzed, and the related conditions for the Hopf bifurcation of the two models are introduced, and some of the theories are also simulated. And the innovations are described as follows: first, the development history of the two types of neural network systems, such as HR and Hopfield, which are studied in this article, are summarized, so that the readers have a more thorough understanding of the two kind of neural network system and the convenience for the follow-up research. Secondly, the article is simple. The relevant theorems and definitions needed for subsequent research are introduced. Then, based on the HR neural network model of Hindmash and Rose and the modeling methods of related literature, a new single time delay neuron network model is established for its addition to the new time delay. The new model is described in detail according to the close relation between the root and the coefficient. There are conditions for the positive equilibrium point, and using the linearization theory and the Hassard method, the condition of the bifurcation of the Hopf bifurcation at the positive equilibrium point, the bifurcation period of the Hopf bifurcation and the decision expression of the bifurcation direction are derived from the standard theory and the central manifold theorem, and the corresponding time history diagram is simulated by using the software software. In the end, considering that the high dimensional nonlinear theory is more practical, the Hopfield, a special four dimensional neural network, is selected for in-depth study. The main innovation is to know the interaction and influence between neurons according to the existing models and related theories, and also in the process of action. There is a lagging phenomenon, so two long connections are added to the original model, and one mutual reverse connection and the corresponding time lag get a new system, which is the second model to be studied in this paper. There are several differences with the research methods of the first model. The difference one is that the system is more special. It can be calculated directly that the model must have a equilibrium point as the original point and do not need to move the non negative equilibrium point again. The difference two is because the system has multiple time delays. It is difficult to do an equivalent transformation according to the relevant theory and transform the original system model into a simple model with only one delay. The same treatment method, theorem and definition of the previous model, the stability of the zero equilibrium point and the existence of the Hopf bifurcation are discussed, and the parameter expression of the bifurcation point of the Hopf is derived, the direction of bifurcation and the periodicity of the moving orbit are obtained, and the stability of the model is also used by the mathematical software. The qualitative theory is tested numerically, which further proves the rationality of the theory.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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本文編號(hào)：1934197