How well does your numerical result agree with the quasi-linear value D = K2/4? (Possibly, you should not take p modulo 2π when looking at its spread)

Assignments week 2 CHAOS, 2021

Hamiltonian Chaos
Write a small report about the assignments listed below. Add (your extensions to) the computer scripts as an appendix.

In this assignment we study two Hamiltonian systems: the periodically kicked rotator, and fluid flow due to vortices which are switched periodically. In both systems, chaos leads to intense mixing in phase space. For the kicked rotator, we can even have an analogy to diffusion, while the vortex flow can mix added tracer particles. In both cases transport is
possible due to nonlinearity and time dependence.

1. Blinking vortex

In this exercise we are going to look at fluid mixing through chaotic advection. We will restrict ourselves to the situation of two-dimensional incompressible flow, where the velocity field satisfies ∇ · v = 0. This is the same as saying that there is a stream function Ψ, with vx = −∂Ψ
∂y (1)
vy = ∂Ψ
∂x .
These two equations are Hamilton’s canonical equations for one degree of freedom. We can identify the stream function Ψ with the Hamiltonian H, x with the momentum p and y with the ‘position’ q.
An example of such a system is the ‘blinking vortex’ flow, described by Aref in [1]. The flow consists of two vortices of equal strength, Ω, one located at (x, y) = (b, 0) and the other located at (x, y) = (−b, 0). Now we will switch the vortices on and off with time period T . This is the ‘blinking’ of the vortices. In this time-periodic flow field, passive tracer particles follow very complicated trajectories. For these particles the equations of motion are:
̇x = − Ω y
x2
s + y2 (2)
̇y = Ω xs
x2
s + y2 ,
with the position xs = x + b for 0 < t < T/2 and xs = x − b when T/2 < t < T . We represent the dynamical state of this system by a stroboscopic map where we register the position of a particle at t = kT , with k an integer number. Such a registration still completely determines the dynamical state of the system. The attractor of this map depends on the dimensionless parameter μ = ΩT/b2 (check it !). We expect that for small μ the map is very close to the integrable case, where we only have fixed points and limit cycles. For larger μ chaotic behavior is expected, and the area occupied by the attractor will grow. In this case there is mixing by chaotic advection.

There is a program showing the tractories of fluid parcels blink w.m, and a template program of the stroboscopic map blink map.m. You can read more about the blinking vortex model in [1] and [2] 1

(1.a) To get started, explore with blink w.m the trajectories of a tracer for Ω = 10, T = 10 and b = 5 (the program asks for T, Ω). Collect some pretty pictures.

(1.b) We just explained how a map can be created by sampling the position of the particles at t = kT . That is done in the template program blink map.m. Create a map for Ω = 10, b = 5 and T = 0.5, starting from a range of initial conditions
(which still has to be programmed).

(1.c) Now we want to vary the strength parameter μ. You can do this by varying T and keep Ω = 10 and b = 5 fixed. Show maps for different values of μ, starting from T = 0.125. Collect some pretty pictures.

(1.d) In the chaotic region, pick many pairs of close points, and see how, on average, their distance increases: mixing ! Try to quantify this mixing process. Get inspiration from Chapter VI of the lecture notes. Or… show the fate of a small
square of initial conditions. (This is an open-ended question…)

2. The periodically kicked rotor.

(From the lecture notes) Consider a a frictionless rotor, which at times t = nτ receives a kick with strength K in the x−direction. We can derive a mapping for this system. The Hamiltonian does not explicitly depend on time; it is
H(pθ, θ, t) = p2
θ
2I + K cos θ ∑
n
δ(t − nτ),
with I the moment of inertia of the rotor. The equations of motion then become
dpθ
dt = −∂H
∂θ = K sin θ ∑
n
δ(t − nτ)
dθ
dt = ∂H
∂pθ
= pθ
I . (3)

1The program integrates the ordinary differential equations Eq. 3, but for this system the trajectories are pieces of circles, so that we could have done the integration analytically. It then amounts to inverting sines and cosines, which is not very insightful.

2In between two kicks pθ is constant, while the angle θ increases with a constant velocity. We can integrate the equations of motion from kick to kick
pn+1 − pn =
∫ (n+1)τ +0
nτ +0
K sin θ ∑
n
δ(t − nτ) dτ = K sin θn+1
θn+1 − θn = pn
τ
I
When we set τ/I = 1, we arrive at the standard map, pn+1 = pn + K sin θn+1 (4)
θn+1 = (θn + pn) mod 2π.

(2.a) A computer program standard.m is provided that lets you play. It provides a beautiful illustration of chaos in Hamiltonian systems. Make pretty pictures of (a) a KAM surface, (b) a Poincar ́e Birkhoff chain.

Chaotic transport

The famous Tokamak fusion plasma reactor, now being built in the south of France, is our hope to solve the energy crisis. I will generate unlimited energy from nuclear fusion using water as a fuel. The tokamak (a torus) is an instance of a Hamiltonian system. The hot plasma is trapped in stable islands. A great concern, however, is the breakup of islands in a chaos transition. The leakage of energy can be understood using the standard map, see the article by Rechester et al. [4].
When K is large there are no visible KAM surfaces present in the standard map, and the entire region of p modulo 2π versus θ is covered by a single chaotic orbit. Also we see that the the change in momentum, according to equation 4, is large. As a result we expect θ to vary wildly and we can treat θn as random and uniformly distributed

In other words, pn increases with random jumps ξn (with average 0), pn+1 = pn + ξn.
After m steps,
pn+m = pn +
m∑
i=1
ξn+i,
with root mean square (rms) distance
∆p2
m = 〈
(pn+m − pn)2〉
=
m∑
i=1
m∑
j=1
〈ξi+nξj+n〉 = m 〈
ξ2〉
= m D, (5)
because ξi and ξj are uncorrelated. Therefore the squared momentum (half the energy)
increases linearly with time: diffusion [3]. When using 〈sin2 θn+1〉 = 1

2 we find for the diffusion coefficient
D = K2
2 ,
see Edward Ott’s nice book on Chaos [3]. When we take initial conditions uniformly spread in θ and p, the momentum distribution function will follow a Gaussian distribution (which must be checked !).

(2.b) Plot p modulo 2π versus θ for orbits with K = 1 and the following five initial conditions:

(θ0, p0) = (π, π/5); (π, 4π/5); (π, 6π/5); (π, 8π/5); (π, 2π).
(2.c) For K = 21 plot the average value of p2 versus iterate number. Average over 100 different initial conditions:

(θ0, p0) = (2nπ/11, 2mπ/11) for n = 1, 2, …, 10 and m = 1, 2, …, 10 and estimate the diffusion coefficient D from equation 5.
How well does your numerical result agree with the quasi-linear value D = K2/4? (Possibly, you should not take p modulo 2π when looking at its spread)

References

[1] Hassan Aref. Stirring by chaotic advection. Journal of Fluid Mechanics, 143(1):1–21,
1984.
[2] G K ́arolyi and T T ́el. Chaotic tracer scattering and fractal basin boundaries in a blinking
vortex-sink system. Physics Reports, 290:125–147, 1997.
[3] Edward Ott. Chaos in Hamiltonian systems. In Chaos in dynamical systems, pages
208–264. 1993.
[4] A. B. Rechester, M. N. Rosenbluth, and R. B. White. Electron heat transport in a
Tokamak with destroyed magnetic surfaces. Phys. Rev. Lett., 40:38, 1978.
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