Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016
Tuin der Lusten (Garden of Earthly Delights)
In all chaos there is a cosmos, in all disorder a secret order (Carl Jung)
Structured - Complex - Chaotic
Chaos has structure
Chaos can move things
Two landmark examples Route to chaos Chaos and structure
Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks
Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks
Mandelbrot set - fractal
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?
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?
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?
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?
Cobweb diagram Sequence of moves : what if n? 1 x3 x2 0 x1 x2 x4 x3 1
Logistic map - dynamics
Bifurcation tree - self-similarity Draw final orbit versus r - period doubling
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?
Mandelbrot set - self-similarity
Logistic map - Mandelbrot and bifurcations
A route to chaos Via period-doubling ever more complex behavior arises, culminating in chaos
Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks
Lorenz attractor
Convective cell - concept
Lorenz system Model for convective cell Sample solution: dx dt dy dt dz dt = 10x + 10y = 28x y xz = 8 3 z + xy
A butterfly in mathematics Striking structure when plotting (x(t), y(t), z(t))
Sensitive dependence Start 10 4 apart
Start 10 4 apart Sensitive dependence Prediction horizon 8
Prediction horizon... Start 10 4 (solid) and 10 7 (dashed) order 1 error after t = 10 (15)
Sensitive orbits - robust structure Analogue: weather (stone) and climate (pond)
Small input may have large consequences...
Outline 1 Route to Chaos 2 Chaos and structure 3 Concluding remarks
Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines
Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines
Concluding remarks chaos is everywhere chaos is fun chaos is relevant chaos sharpens mathematics mathematics clarifies chaos chaos is universal - connects the disciplines