非線性學(xué)簡(jiǎn)介-中國(guó)物理學(xué)會(huì)XXXX年秋季學(xué)術(shù)會(huì)議ppt課件

1、非線性學(xué)簡(jiǎn)介1.引入中文2.舉例英文3.前景英文.Introduction非線性學(xué)是一門研討非線性系統(tǒng)的共性，探求事物復(fù)雜性的新學(xué)科(science of complexity)。所謂非線性是相對(duì)線性而言的。什么是非線性學(xué)線性是指量與量之間的正比關(guān)系，在平面直角坐標(biāo)系統(tǒng)中，表現(xiàn)為直線或曲線。在線性系統(tǒng)中，分量之和等于總量f(x+y) = f(x)+f(y) and f(ax) = af(x), 描畫線性系統(tǒng)的方程遵照疊加原理(principle of superposition)，即方程的不同解加起來(lái)依然是解。而非線性那么剛好相反，分量之和不等于總量，不遵照疊加原理。. 非線性的物理景象普通具

2、有這么幾個(gè)特征非線性景象在時(shí)空中表現(xiàn)為從規(guī)那么運(yùn)動(dòng)向不規(guī)那么運(yùn)動(dòng)的轉(zhuǎn)化和騰躍非線性系統(tǒng)中某個(gè)分量極微小的變化，可以引起整個(gè)系統(tǒng)運(yùn)動(dòng)方式的性質(zhì)改動(dòng)(參見(jiàn)蝴蝶效應(yīng))非線性系統(tǒng)對(duì)外界擾動(dòng)會(huì)作出與外界擾動(dòng)要素截然不同的呼應(yīng)非線性作用普通會(huì)導(dǎo)致空間規(guī)整性構(gòu)造的構(gòu)成和維持 .19世紀(jì)末龐加萊(H.Poincare)正是在總結(jié)整個(gè)世紀(jì)這方面進(jìn)展的根底上，提出不少新的實(shí)際和方法，當(dāng)前非線性科學(xué)中的很多概念和思想，都根源于龐加萊。 非線性科學(xué)中，那些可以有定量分析、準(zhǔn)確計(jì)算、數(shù)學(xué)實(shí)際或?qū)嶒?yàn)研討的部分，普通以為可以歸為以下三種：孤立波(soliton)，混沌(chaos)，分形(fractal).孤立波，以及相應(yīng)

3、的孤立子的研討，是這三者中開(kāi)展較早的一個(gè)。當(dāng)然它的發(fā)現(xiàn)可以追溯到十九世紀(jì)羅素騎馬時(shí)在一個(gè)河道中看到的一個(gè)孤立波，他騎著馬跟著這個(gè)波，奇異的是它直到3-4英里以后才破碎。水波的第一個(gè)孤立波的解的發(fā)現(xiàn)也是遲至上世紀(jì)六十年代由克魯斯卡爾 (Kruskal) 等人作出的。孤立波或孤立子從那以后就幾乎成了一個(gè)獨(dú)立學(xué)科。在很多情況下，孤立子的解看起來(lái)很難找到，但在一些簡(jiǎn)單的模型里可以用簡(jiǎn)單的方法找到。到今天，除了沿它本身體系開(kāi)展外，由于它在數(shù)學(xué)處置上已獲得不少閱歷，我們指望從而得到了解其他非線性景象中圖型構(gòu)成的機(jī)理。比如，有空間傳播性能的波形不變的非線性景象，可以以為是系統(tǒng)中由于自組織而“降維，在數(shù)學(xué)上和

4、非線性振動(dòng)中的所謂同宿解有關(guān)。對(duì)其他非線性景象的了解能夠從孤立波已有成果得到自創(chuàng)。 混沌，指一種貌似無(wú)規(guī)的運(yùn)動(dòng)，但支配它這種運(yùn)動(dòng)的規(guī)律卻可用確定型的方程來(lái)描畫。上面提到的龐加萊在總結(jié)天膂力學(xué)中的問(wèn)題時(shí)，曾經(jīng)對(duì)這種景象有了認(rèn)識(shí)。到20世紀(jì)50年代，有些物理學(xué)家(如玻恩(M.Born)也已明確知道經(jīng)典力學(xué)中會(huì)有長(zhǎng)期動(dòng)態(tài)的不可預(yù)測(cè)性。但混沌景象和實(shí)際開(kāi)場(chǎng)遭到注重，普通以為契機(jī)于60年代兩件事。一是羅侖茲(E.Lorenz)在天氣預(yù)告方程的研討中發(fā)現(xiàn)，雖然描畫用的方程是確定性的，天氣長(zhǎng)期動(dòng)態(tài)卻是不可預(yù)測(cè)的。另一是，幾位數(shù)學(xué)家證明了有關(guān)經(jīng)典力學(xué)動(dòng)態(tài)的一個(gè)定理，即如今按他們的姓稱謂的卡姆(KAM)實(shí)際。這

5、兩件事也分別代表混沌實(shí)際兩類對(duì)象和兩種方法：羅侖茲的對(duì)象是耗散系統(tǒng)(這類系統(tǒng)和周圍環(huán)境有聯(lián)絡(luò)、有交往，它們?cè)谧匀缓凸こ讨卸加?，而卡姆的對(duì)象是保守系統(tǒng)(當(dāng)作是孤立的、封鎖的，它們?cè)谔祗w研討和統(tǒng)計(jì)物理中常見(jiàn))。羅侖茲依托的是數(shù)值計(jì)算，卡姆用的是嚴(yán)厲數(shù)學(xué)推理，這兩種方法在混沌實(shí)際研討里都是必不可少的。當(dāng)前混沌實(shí)際所面臨的數(shù)學(xué)情況比分形實(shí)際好些，但不如孤立波?，F(xiàn)有的數(shù)學(xué)有的對(duì)混沌實(shí)際很起作用，也有些問(wèn)題那么還沒(méi)有找到稱手的數(shù)學(xué)工具。 .分形和不規(guī)那么外形的幾何有關(guān)。人們?cè)缇褪炝?xí)從規(guī)那么的實(shí)物籠統(tǒng)出諸如圓、直線、平面等幾何概念，芒德波羅(B.B.Mandelbrot)那么對(duì)曲曲彎彎的海岸線、棉絮團(tuán)似的

6、云煙找到適宜的幾何學(xué)描畫方法分形。分形實(shí)際出現(xiàn)較晚，它的數(shù)學(xué)預(yù)備不象孤立波那樣充分，目前它的數(shù)學(xué)實(shí)際和實(shí)踐運(yùn)用之間間隔還較大，有些數(shù)學(xué)概念還得從頭重新建立。比如，微積分里導(dǎo)數(shù)是和光滑曲線的斜率相聯(lián)絡(luò)的，對(duì)于曲曲彎彎海岸線那樣的曲線，導(dǎo)數(shù)又怎樣定義?假設(shè)象微分積分那樣的操作都沒(méi)有，那就很難做進(jìn)一步的定量的研討。分形數(shù)學(xué)和分形物理的結(jié)合還剛開(kāi)場(chǎng)。以上三項(xiàng)內(nèi)容是彼此聯(lián)絡(luò)著的，也還和其他問(wèn)題有關(guān)。當(dāng)一個(gè)系統(tǒng)或事物里有可調(diào)的參量( 設(shè)參量本身不參與隨時(shí)間變化)，參量不同會(huì)引起系統(tǒng)長(zhǎng)期動(dòng)態(tài)發(fā)生什么根本的(定性)變化，這是“分岔實(shí)際所關(guān)懷的問(wèn)題。當(dāng)參量變化跨越某些臨界值(叫做分岔點(diǎn))，系統(tǒng)將有根本的轉(zhuǎn)變，比

7、如孤立波失穩(wěn)了，或者一種分形構(gòu)造變化了，混沌過(guò)程變成周期振蕩了，等等。再有，假設(shè)在一系統(tǒng)或事物的演化中，從時(shí)間過(guò)程看有混沌，而在空間分布上又有變化著的分形圖型，就得時(shí)空聯(lián)絡(luò)起來(lái)研討圖型的動(dòng)力學(xué)。正是本著這樣的觀念，在非線性科學(xué)這個(gè)艱苦工程里的各個(gè)課題，是既有分工又有聯(lián)絡(luò)。.ExamplesWhat is the butterfly effect? The butterfly effect is a term used to describe the principles of nonlinearity and sensitivity to initial conditions, which h

8、old that a nonlinear equation can have solutions that are irregular. The irregularity results in small changes being amplified by the nonlinear nature of the system. This means that if the initial state of the nonlinear system is changed only slightly, one cannot predict the difference in how each s

9、ystem will evolve over time. One often-cited example of the effects of nonlinearity and sensitivity to initial conditions was given by the meterologist, Ed Lorenz. He explained, mathematically, why predicting the weather with precision is impossible. Lorenz demonstrated that two virtually identical

10、weather systems will behave differently over time due to their complex, nonlinear nature and due to inputs from the environment that are infinitely small. He suggested, somewhat tongue-in-cheek, that even the flapping wings of a butterfly could result in a tornado because of nonlinear processes at w

11、ork even with the smallest factors causing the weather. .What is Nonlinear Acoustics/Elasticity Strike a bell, and the bell rings at its resonance modes. Strike it harder and the bell rings at the same tone, only louder. Now imagine a small crack in the bell, perhaps invisible to the eye. We strike

12、the bell gently and it rings normally. Striking it harder we find, to our surprise, that the tone drops in frequency ever so slightly. Striking it even harder, the tone drops farther down in frequency. The frequency shift is a manifestation of nonlinearity due to the presence of the crack. Figure 1

13、illustrates how the bell responds elastically linearly when undamaged, but elastically nonlinearly when damaged. The bell behaves in an expected manner when intact (Figure 1a). Ringing the bell with a hammer excites its resonance modes, giving rise to a frequency spectrum in which only the modal fre

14、quencies are present. However, if the bell has even a very small crack present, the resonance frequencies depend on how hard the bell is struck (Figure 1b). This is a nonlinear effect: a change in wave frequency with wave amplitude. .Figure 1: Linear and nonlinear resonance in a bell.Figure 2: Linea

15、r and nonlinear harmonic generation and frequency Modulation in a bell. This example is taken a step further in Figure 2 for the sake of illustrating additional manifestations of nonlinearity. For instance, we input signals with frequencies of 440Hz and 8000Hz into the undamaged bell using an audio

16、speaker (these are arbitrarily chosen frequencies and are not crucial to the general result). Not suprisingly, the bell will ring at the two input frequencies (Figure 2a). If we input the two tones into the bell when a small crack is present, interesting things happen again. We find that, not only d

17、oes the bell ring at 440 and 8000 Hz, but other frequencies abound, as illustrated in Figure 2b. We observe spectral components at two times, three times and four times each input frequency (880, 1320, and 1740 Hz; and 16000, 24000, and 32000 Hz, respectively). In addition, we detect the sum and dif

18、ference frequencies between the 440 and 8000 Hz: 8000+/-440 Hz (These frequencies are called sidebands.). The resonance peak change with amplitude, and the appearance of new frequencies inside the material, are not expected results! They are the result of nonlinear interaction of the sound in the damaged bell. The nonlinearity due to the presence of the crack(s) is (are) an extremely sensitive indicator of the presence of damage. The undamaged portion of the sample produces nearly zero nonlinear effect.