Chaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB

Transcription

1 Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB

2 Examples of chaotic systems: the double pendulum? θ 1 θ θ 2

3 Examples of chaotic systems: the double pendulum

4 Examples of chaotic systems: the double pendulum Strogatz in action...

5 Examples of chaotic systems: the chaotic waterwheel

6 Examples of chaotic systems: the chaotic waterwheel

7 Examples of chaotic systems: the chaotic waterwheel = angle in the lab frame!(t) = angular velocity of the wheel m(,t) = mass distribution of the wheel Q( ) =inflow r = radius of the wheel K = leakage rate = rotational damping rate I = moment of inertia of the wheel

8 The equations of motion for the chaotic waterwheel For details of derivation: see book Conservation of mass m t = Q Km! m Torque balance I! t =! + gr Z 2 0 m(,t)sin d

9 Reducing the chaotic waterwheel equations m t = Q Km! m I! Z 2 t =! + gr m(,t)sin d 0 m(,t)= Q( ) = 1X [a n (t)sinn + b n (t) cos n ] n=0 1X n=0 q n cos n

10 Reducing the chaotic waterwheel equations ȧ n = n!b n Ka n ḃ n = n!a n Kb n + q n ȧ 1 =!b 1 Ka 1 ḃ 1 =!a 1 Kb 1 + q 1! =(! + gra 1 )/I ẋ = (y x) ẏ = rx y xz ż = xy bz ~ Lorenz equations

11 The waterwheel equations: fixed points 0=!b 1 Ka 1 0=!a 1 Kb 1 + q 1 0=(! + gra 1 )/I! =0 b 1 = q 1 /K! 2 = grq 1 b 1 = K gr K 2 (with grq 1 K 2 > 1) dimensionless number Rayleigh number no rotation CW/CCW rotation

12 The Lorenz equations ẋ = (y x) ẏ = rx y xz ż = xy bz r: Rayleigh number : Prandtl number b: aspect ratio - two quadratic nonlinearities - symmetric under (x,y) -> (-x,-y) - dissipative: volumes in phase space contract under the flow (for details: see book) J. Atmos. Sci 20 (1963), pp

13 The Lorenz equations: fixed points ẋ = (y x) ẏ = rx y xz ż = xy bz (x, y, z) =(0, 0, 0) (for all r) Conduction x 2 = y 2 = b(r z = r 1 1) (with r>1) = 10 b =8/3 Convection cells born at supercritical pitchfork bifurcation, C + and C -

14 The Lorenz equations: fixed points and local stability ẋ = (y x) ẏ = rx y ż = bz = 1 < 0 = (1 r) 2 4 =( 1) 2 +4 r>0 Linearization - r < 1: origin = stable node (also globally stable - see book) - r > 1: origin = saddle point + b +3 - C + and C - linearly stable for 1 <r<r H = b 1 (r H for standard parameters)

15 The Lorenz equations: fixed points and local stability What happens after subcritical Hopf? - no stable cycles - no trajectories to infinity -> Strange attractor? - no torus

16 The Lorenz equations: exercise on computer - explore numerically the time dynamics when varying r (r = 0.5, 10, 21, 24.1, 28, 100, 350) - plot phase space portraits - check what happens small changes in initial conditions - calculate how such small changes evolve in time

17 The Lorenz equations: exercise on computer - explore numerically the time dynamics when varying r

18 The Lorenz equations: exercise on computer - plot phase space portraits

19 The Lorenz equations: exercise on computer - check what happens to small changes in initial conditions (t) 0 e t

20 The Lorenz equations: exercise on computer - check what happens to small changes in initial conditions (t) 0 e t t horizon 1 ln a 0

21 The Lorenz equation: exercise on computer (e.g. r = 0.5, 10, 21, 24.1, 28, 100, 350)

22 How can we define chaos? Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions

23 Can we use a 1D map to gain understanding into chaos?

24 Fixed point and limit cycles are unstable in the map - stability

25 Route to chaos in the 1D logistic map: exercise on computer x n+1 = rx n (1 x n ) x n 2 [0, 1] r 2 [0, 4] - explore numerically the time dynamics when varying r - interpret with cobwebb diagrams - plot orbit diagram (extrema in function of r) R. May (1976), Nature 261, 459

26 Route to chaos in the 1D logistic map: period doubling

27 Route to chaos in the 1D logistic map: period doubling r n r n 1 r n+1 r n! 4.669

28 The logistic map: stability fixed points and cycles x n+1 = rx n (1 x n ) - via linearization (try at home) - using the cobwebb diagram 1. origin globally stable: 2. stable non-zero fixed point: x = 0; r<1 3. oscillations and period doubling : x 1 =1 r ;1<r<3 f 0 (x )= 1; r =3

29 The logistic map: stability fixed points and cycles 3. oscillations and period doubling : - period 2 cycle exists for r > 3 f(f(x)) = f 2 (x) =x f 2 (x) =r(rx(1 x))(1 (rx(1 x))) = x p, q = r +1± p (r 3)(r + 1) 2r

30 The logistic map: stability fixed points and cycles 3. oscillations and period doubling : - stability of period 2 cycle? d dx (f(f(x))) x=p = f 0 (f(p))f 0 (p) = f 0 (q)f 0 (p) = r(1 2q)r(1 2p) =4+2r r 2 stable when abs < 1: p r 2 =1+ 6=3.4495

31 Universality Exercise on the computer: explore the sine map (also unimodal): x n+1 = rsin( x n ) r 2 [0, 1] x 2 [0, 1] - qualitative correspondence: orbit diagram - quantitative correspondence: Feigenbaum r n r n 1 r n+1 r n! 4.669

32 Universality: Feigenbaum n n+1! d n d n+1!