Een vlinder in de wiskunde: over chaos en structuur

Transcription

1 Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016

2 Tuin der Lusten (Garden of Earthly Delights)

3

4

5

6

7 In all chaos there is a cosmos, in all disorder a secret order (Carl Jung)

8 Structured - Complex - Chaotic

9 Chaos has structure

10 Chaos can move things

11 Two landmark examples Route to chaos Chaos and structure

12 Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks

13 Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks

14 Mandelbrot set - fractal

15 Mapping as a model - logistic map x f y Generate sequence: linear Orbit x n+1 = f(x n ) = rx n ; x 0 = 1 x n = r n If r < 1 then x n 0 and if r > 1 then x n Nonlinear x n+1 = f(x) = rx n (1 x n ) ; Dynamics?

16 Mapping as a model - logistic map x f y Generate sequence: linear Orbit x n+1 = f(x n ) = rx n ; x 0 = 1 x n = r n If r < 1 then x n 0 and if r > 1 then x n Nonlinear x n+1 = f(x) = rx n (1 x n ) ; Dynamics?

17 Mapping as a model - logistic map x f y Generate sequence: linear Orbit x n+1 = f(x n ) = rx n ; x 0 = 1 x n = r n If r < 1 then x n 0 and if r > 1 then x n Nonlinear x n+1 = f(x) = rx n (1 x n ) ; Dynamics?

18 Mapping as a model - logistic map x f y Generate sequence: linear Orbit x n+1 = f(x n ) = rx n ; x 0 = 1 x n = r n If r < 1 then x n 0 and if r > 1 then x n Nonlinear x n+1 = f(x) = rx n (1 x n ) ; Dynamics?

19 Cobweb diagram Sequence of moves : what if n? 1 x3 x2 0 x1 x2 x4 x3 1

20 Logistic map - dynamics

21 Bifurcation tree - self-similarity Draw final orbit versus r - period doubling

22 Mandelbrot set - bounded orbits? Consider z z 2 + c Orbit of 0 0 c c 2 + c (c 2 + c) 2 + c... For which c is the orbit of 0 a bounded orbit?

23 Mandelbrot set - self-similarity

24 Logistic map - Mandelbrot and bifurcations

25 A route to chaos Via period-doubling ever more complex behavior arises, culminating in chaos

26 Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks

27 Lorenz attractor

28 Convective cell - concept

29

30 Lorenz system Model for convective cell Sample solution: dx dt dy dt dz dt = 10x + 10y = 28x y xz = 8 3 z + xy

31 A butterfly in mathematics Striking structure when plotting (x(t), y(t), z(t))

32 Sensitive dependence Start 10 4 apart

33 Start 10 4 apart Sensitive dependence Prediction horizon 8

34 Prediction horizon... Start 10 4 (solid) and 10 7 (dashed) order 1 error after t = 10 (15)

35 Sensitive orbits - robust structure Analogue: weather (stone) and climate (pond)

36 Small input may have large consequences...

37

38 Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks

39 Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines

40 Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines

41 Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines

42