Reactive Fluid Dynamics グローバル COE 環境共生 安全システムデザインの先導拠点 複雑システムのデザイン体系 第 2 回 力学系の非線形力学 理工学研究科, 開放環境科学専攻 教授植田利久 2009 年 4 月 16 日. Keio University. No.

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1 グローバル COE 環境共生 安全システムデザインの先導拠点 複雑システムのデザイン体系 第 回 力学系の非線形力学 理工学研究科, 開放環境科学専攻 教授植田利久 9 年 4 月 16 日 No.

2 contents 1. What is a non-linear? 1.1 Linear unction and non-linear unction 1. Non-linear variation. Mechanical engineering: historical viewpoint 3. Basic non-linear dnamics 3.1 Chaos, the deinition 3. Phase space, phase space diagram, attractor, dimension and Poincare section 3..1 Phase space and Phase space diagram 3.. Attractor 3..3 Dimension 3..4 Poincare Section 3..5 The logistic map 3..6 Route to chaos 3..7 Stretching and olding No.1

3 Reerences Baker,G.L. and Gollub,J.P., Chaotic Dnamics, an introduction, Cambridge, 1994 Devane,R.L., 後藤憲一ほか訳, 新訂版カオス力学系入門第 版, 共立出版, 3 合原一幸編, カオス時系列解析の基礎と応用, 産業図書, 長島弘幸, 馬場良和共著, カオス入門現象の解析と数理, 培風館, 1999 山口昌哉, カオス入門, カオス全書, 朝倉書店, 1998 No.

4 Deinition o the linear unction 1 What is a non-linear? No.3 a a + + a a a a a a a a a Linear unction and non-linear umction Linear unction Non-linear unction

5 1 What is a non-linear? 1. Non-linear variation I No ^ ^ No.4

6 1 What is a non-linear? 1. Non-linear variation II No a No.5

7 Mechanical Engineering: historical viewpoint Most o phenomena are non-linear In th centur The non-linear phenomena is treated assuming local linearit. e e In 1st centur The non-linear phenomena is treated as itsel constructing new non-linear dnamics e - 6

8 3 Basic non-linear dnamics 3.1 Chaos, the deinition Oord English dictionar describes: chaos [keias] is a state o complete conusion and disorder. The chaos in the ield o engineering: is the deterministic chaos The complicated behavior o dnamical sstem arising rom a strictl deterministic time evolution without an source o noise or eternal stochasticit. In other words, the chaotic motion is not stochastic but deterministic. Thus, the chaotic motion can be traced and predicted no matter how the motion is complicated. 7

9 3 Basic non-linear dnamics 3. Phase space, phase space diagram, attractor, dimension, and Poincaré section 3..1 Phase space and Phase space diagram phase space: characterie dnamic sstems phase space diagram: The trajector o the behavior in the phase space Phsical space a Phase space t v 8

10 3 Basic non-linear dnamics 3.. Attractor attractor: A phase space diagram that shows universal characteristics o the behavior independent rom initial conditions. Periodic Chaotic Stead b t Bi-Periodic d t Random a t c t e t a point attractor b limit ccle c torus attractor d strange attractor e random motion 9

11 3 Basic non-linear dnamics 3..3 Dimension Dimension is understood as a number o reedom In other words, the number o coodinate point: ero dimension straight line: one dimension Plane: two dimension space: three dimension Thus, the number o dimension is integer 1

12 Basic non-linear dnamics 3..3 Dimension The question was arisen. An arbitrar point in the n dimensional space can be epressed b one real number when we cover the space b one line, like a Peano curve It indicates that n dimensional space can be considered to be one dimensional rom the view point o reedom 11 Peano curve

13 3 Basic non-linear dnamics 3..3 Dimension The capacit dimension is deined as, log N ε dc lim ε log1/ ε here, ε is the sie o bo to cover the igure and N is the number o bo to cover the igure. 1

14 3 Basic non-linear dnamics 3..3 Dimension The capacit dimension o two-dimensional square is calculated two as ollows. dc log N ε lim ε log1/ ε n ε N 11/ 1 1 1/1/ 1 1 1/41/ 4 i 1/ i i n 1/ n n dc lim ε log N ε log1/ ε lim ε n n log log 1 13

15 3 Basic non-linear dnamics 3..3 Dimension n ε N 11/ 1 dc lim ε 1 1/1/ 1 4 log N ε log1/ ε lim ε nlog nlog 1/41/ 16 4 i 1/ i i n 1/ n n n ε N 11/ 1 dc lim ε 1 1/1/ log N ε log1/ ε lim ε 1/41/ n log nlog 3 i 1/ i 3i n 1/ n 3n 14

16 3 Basic non-linear dnamics 3..3 Dimension Cantor set n ε N 11/ /31/ /91/ /71/ i 1/3 i i n 1/3 n n dc lim ε log N ε log1/ ε lim ε nlog nlog 3 < 1 15

17 3 Basic non-linear dnamics 3..3 Dimension The important point on the dimension in the chaotic dnamics is that the chaotic motion have non-integer dimension, and then, the dimension o the strange attractor is non-integer. 16

18 3 Basic non-linear dnamics 3..4 Poincaré Section A Poincaré section is a device invented b Henri Poincaré as a means o simpliing phase space diagram o complicated sstems. It is constructed b viewing the phase space diagram stroboscopicall. The Poincaré section allows a sstematic reduction in compleit o problems b means o a reduction in the number o dimension. The Poincaré section can be constructed b deining a surace o section in the phase space. 17

19 3 Basic non-linear dnamics 3..4 Poincaré Section Σ Σ Σ Σ Σ a point attractor b limit ccle c torus attractor d strange attractor e random motion Point attractor Limit ccle Torus Strange attractor Random motion Attractor 1 Non-integer 3 Poincaré map - 1 Non-integer 18

20 3 Basic non-linear dnamics 3..5 The logistic map 1 n+ 1 µ n n 1.75 μ μ. n+1.5 n n n μ<1 go to 1<μ<3 go to 1-1/ μ 19

21 3 Basic non-linear dnamics 3..5 The logistic map 1 n+ 1 µ n n 1.75 μ μ3.1 n+1.5 n n n 3<μ<1+ 6 period-

22 3 Basic non-linear dnamics 3..5 The logistic map 1 n+ 1 µ n n 1.75 μ μ3.5 n+1.5 n n n 1+ 6<μ period-4 1

23 3 Basic non-linear dnamics 3..5 The logistic map 1 n+ 1 µ n n 1.75 μ μ3.8 n+1.5 n n 1+ 6<μ period n <μ chaotic state Period doubling mechanism, biurcation, is one route to chaos

24 3 Basic non-linear dnamics 3..5 The logistic map 1 n+ 1 µ n n Biurcation diagram 3

25 3 Basic non-linear dnamics 3..6 Route to chaos attractor o period doubling is show below:. Periodic Chaotic Stead b t Bi-Periodic d t Random a t c t e t a point attractor b limit ccle c torus attractor d strange attractor e random motion 4

26 3 Basic non-linear dnamics 3..7 Stretching and olding to realie instead o, the stretching and olding is important luid element stretching olding n 5

27 3 Basic non-linear dnamics 3..7 Stretching and olding to realie instead o, the stretching and olding is important D1mm Ater 1times stretching and olding, the number o line becomes 14. The line length becomes 1.4m, while.1m o linear stretching linear stretching Stretching and olding Then, the width between lines becomes onl 1/14 mm 1μm, while leaving 1mm b the linear stretching More stretching in less area 6