BlinkingRotlet

Systems / ODEs / Climate & geophysics

A passive tracer stirred by two off-centre rotlets that alternate on and off, folding and stretching material lines into chaotic Stokes-flow advection.

continuous · ODE3 dimensionschaotic advection

phase portrait

Definition

\[ \begin{aligned} \dot{r} &= \sigma \left(- b \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(- \frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right)}{\left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} + \frac{1}{b^{2} + 2 b r \cos{\left(\theta \right)} + r^{2}} - \frac{1}{a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}}\right) \sin{\left(\theta \right)} + b \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \left(- \frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right)}{\left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} + \frac{1}{b^{2} - 2 b r \cos{\left(\theta \right)} + r^{2}} - \frac{1}{a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}}\right) \sin{\left(\theta \right)}\right) \\ \dot{theta} &= \frac{\sigma \left(\left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(\frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right) \left(b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}\right)}{\left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} - \frac{b \cos{\left(\theta \right)} + r}{b^{2} + 2 b r \cos{\left(\theta \right)} + r^{2}} + \frac{b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}}{a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}} + \frac{bc r \left(a^{2} + b^{2} - \frac{2 b^{2} r^{2}}{a^{2}}\right)}{a^{2} \left(a^{2} + 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)}\right) + \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \left(\frac{bc \left(1 - \frac{r^{2}}{a^{2}}\right) \left(a^{2} - \frac{b^{2} r^{2}}{a^{2}}\right) \left(- b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}\right)}{\left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)^{2}} - \frac{- b \cos{\left(\theta \right)} + r}{b^{2} - 2 b r \cos{\left(\theta \right)} + r^{2}} + \frac{- b \cos{\left(\theta \right)} + \frac{b^{2} r}{a^{2}}}{a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}} + \frac{bc r \left(a^{2} + b^{2} - \frac{2 b^{2} r^{2}}{a^{2}}\right)}{a^{2} \left(a^{2} - 2 b r \cos{\left(\theta \right)} + \frac{b^{2} r^{2}}{a^{2}}\right)}\right)\right)}{r} \\ \dot{t} &= 1 \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	cell radius
b	0.529883	rotlet offset radius
bc	1	rotlet strength
sigma	-1	rotlet circulation sign/scale
tau	3	blinking period

State variables: r, theta, t

Properties

Lyapunov spectrum

$+0.01541,\; -0.01421,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

n/a — too large to display — pathological rational divergence

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.BlinkingRotlet()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Meleshko & Aref (1996), Phys. Fluids 8, 3215-3217

BibTeX

@misc{blinkingrotlet,
  title = {BlinkingRotlet system},
  note = {Meleshko & Aref (1996), Phys. Fluids 8, 3215-3217}
}