Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest

Transcription

1 Creating and Analyzing Chaotic Attractors Using Mathematica Presented at the 2013 MAA MathFest Ulrich Hoensch 1 Rocky Mountain College Billings, Montana hoenschu@rocky.edu Saturday, August 3, Travel funded by the Montana Space Grant Consortium and Montana NASA EPSCoR.

2 The Undamped Pendulum Consider a pendulum with an object of mass m attached to a rigid swing arm of length l. Θ Θ mg

3 The Undamped Pendulum Let θ be the angle (in radians) that the arm of the pendulum makes with the downward vertical. Then Newton s Second Law gives θ + g l sin θ = 0. (1) Letting ψ = θ, we obtain the nonlinear first order system θ = ψ (2) ψ = g l sin θ. This model excludes any frictional forces. Thus, we have conservation of mechanical energy, and can think of (2) as a Hamiltonian system. The Hamiltonian (total energy function) can be chosen to be H(θ, ψ) = ψ2 2 g cos θ. (3) l

4 The Damped Pendulum If we add in frictional forces that are proportional to the angular velocity dθ/dt, we get the equation d 2 θ dt 2 + r dθ dt + g sin θ = 0, (4) l Adding external sinusoidal forcing with amplitude A and circuluar frequency ω yields the differential equation d 2 θ dt 2 + r dθ dt + g sin θ = A sin(ωt), (5) l

5 The Damped Pendulum If g/l = 1, r = 0.2, A = 0.5 and ω = 2π, then a typical orbit looks like this: Τ Θ 1 0 Ψ 1 2

6 The Forced Damped Pendulum The figure below shows the same orbit when t = 0, 2π, 4π,..., 20π in the (θ, ψ)-plane only. It appears that the orbit approaches the fixed point S ( 1, 1.2). 1 Ψ

7 The Poincaré map Evaluating the orbit (θ(t), ψ(t)) at times t = 0, 2π, 4π,... defines a Poincaré map S(θ(t), ψ(t)) = (θ(t + 2π), ψ(t + 2π)). (6) The Poincaré map can be computed numerically by integrating (5). The following code defines the Poincaré map and orbit in Mathematica. g 1; l 1; r 0.2; Ω 1; Poincare Θ_, Ψ_, A_ : Flatten Evaluate Mod x t, 2 Pi, Pi, y t. NDSolve x' t y t, y' t g l Sin x t r y t A Sin Ω t, x 0 Θ, y 0 Ψ, x t, y t, t, 0, 2 Pi Ω. t 2 Pi Ω ; PList Θ_, Ψ_, A_, nmin_, nmax_ : Module M, n, M 0 : Θ, Ψ ; M n_ : M n Poincare M n 1, A ; Table M n, n, nmin, nmax ;

8 The Forced Damped Pendulum Now, suppose g/l = 1, r = 0.2, A = 2.2 and ω = 2π in (5). If we plot the positions at t = 2πn for 100 n 500, we obtain the following picture Ψ

9 The Forced Damped Pendulum Here is a picture using 250,000 points. The set produced is a chaotic attractor.

10 The Experiment of Moon and Holmes A steel beam (light blue) which is suspended equidistantly between two magnets of equal strength (red).

11 The Experiment of Moon and Holmes If this apparatus is shaken horizontally with fixed amplitude γ and period 2π, the horizontal strain on the beam can be modeled using the forced Duffing equation d 2 y dt 2 + ν dy dt + y 3 y = γ sin t, (7) where ν > 0 and γ 0 are parameters. If ν = 0.05 and γ = 2.8, we again obtain a chaotic attractor (shown on the next slide, using 250,000 points).

12 The Experiment of Moon and Holmes

13 Building Your Own Attractor Note that if ν = γ = 0 in (7), we obtain the Hamiltonian system with H(y, ẏ) = (ẏ)2 2 + y 4 4 y 2 2. (8) The potential energy function y y 4 /2 y 2 /2 confirms the bi-stable attractor when ν > 0 and γ = 0: y

14 Building Your Own Attractor We can create a tri-stable attractor by e.g. integrating the function y y(y 2 1)(y 2 4) to obtain the potential energy function y 2y 2 5y 4 /4 + y 6 / y

15 Building Your Own Attractor The corresponding differential equation that includes friction and external forcing is d 2 y dt 2 + ν dy dt + y(y 2 1)(y 2 4) = γ sin t. (9) If ν = 0.07 and γ = 9.9, we obtain the following chaotic attractor.

16 Dimension of the Chaotic Attractor of the Pendulum Let Ω be the chaotic attractor of the forced damped pendulum as above. To numerically approximate its box-counting dimension, we need to find the number of boxes needed to cover Ω at a given scale ɛ. If ɛ = 1, N(ɛ) = 22:

17 Dimension of the Chaotic Attractor of the Pendulum If ɛ = 1/2, N(ɛ) = 59:

18 Dimension of the Chaotic Attractor of the Pendulum If ɛ = 1/4, N(ɛ) = 164:

19 Dimension of the Chaotic Attractor of the Pendulum If ɛ = 1/8, N(ɛ) = 470:

20 Dimension of the Chaotic Attractor of the Pendulum Using the finer scales of ɛ = 1/10, 1/20, 1/30, 1/40, 1/50, we obtain the following data (using 250,000 points for the attractor): ɛ N(ɛ) log(1/ɛ) log(n(ɛ)) 1/ / / / /

21 Dimension of the Chaotic Attractor of the Pendulum The log(1/ɛ)-log(n(ɛ)) graph shows a slope of d 1.43 for the regression line (r 2 = 0.999). Thus, dim(ω) Log N Ε Log 1 Ε

22 Dimension of the Chaotic Attractor of the Pendulum The following Mathematica code allows the computation of the box-counting dimension of a list of points at the given scales. 8-Pi, Pi, scale <, NBoxes@points_, scale_d := 8-5, 1, scale <DD, _? Hð > 0 &LD; scales = 81 10, 1 20, 1 30, 1 40, 1 50<; Map@NBoxes@list, ð D &, scalesd; data = ð D, ð DD< &, scalesd; model = x, xd Out[8]= FittedModel B x F

23 References [1] Blanchard, P., Devaney, R., Hall, G. Differential Equations, Fourth Edition (2012), Brooks/Cole Cengage Learning. [2] Hoensch, U. Differential Equations and Applications Using Mathematica (2012), available as PDF download at [3] Lynch, S. Dynamical Systems with Applications Using Mathematica (2007), Birkhäuser Boston. [4] Ott, E. Chaos in Dynamical Systems, Second Edition (2002), Cambridge University Press.