InteriorSquirmer

Systems / ODEs / Climate & geophysics

The unsteady interior Stokes flow of an oscillating cylindrical squirmer, advecting tracers chaotically through a five-mode surface actuation.

continuous · ODE3 dimensionschaotic advection

phase portrait

Definition

\[ \begin{aligned} \dot{r} &= 5 r^{4} \left(r^{2} - 1\right) \left(0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \cos{\left(5 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \sin{\left(5 \theta \right)}\right) + 4 r^{3} \left(r^{2} - 1\right) \left(0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \cos{\left(4 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \sin{\left(4 \theta \right)}\right) + 3 r^{2} \left(r^{2} - 1\right) \left(0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \cos{\left(3 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \sin{\left(3 \theta \right)}\right) + 2 r \left(r^{2} - 1\right) \left(0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \cos{\left(2 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \sin{\left(2 \theta \right)}\right) + \left(r^{2} - 1\right) \left(0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \cos{\left(\theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \sin{\left(\theta \right)}\right) \\ \dot{theta} &= \frac{r^{5} \left(2 r + \frac{5 \left(r^{2} - 1\right)}{r}\right) \left(- 0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \sin{\left(5 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \cos{\left(5 \theta \right)}\right) + r^{4} \left(2 r + \frac{4 \left(r^{2} - 1\right)}{r}\right) \left(- 0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \sin{\left(4 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \cos{\left(4 \theta \right)}\right) + r^{3} \left(2 r + \frac{3 \left(r^{2} - 1\right)}{r}\right) \left(- 0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \sin{\left(3 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \cos{\left(3 \theta \right)}\right) + r^{2} \left(2 r + \frac{2 \left(r^{2} - 1\right)}{r}\right) \left(- 0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \sin{\left(2 \theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \cos{\left(2 \theta \right)}\right) + r \left(2 r + \frac{r^{2} - 1}{r}\right) \left(- 0.5 \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \sin{\left(\theta \right)} + 0.5 \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \cos{\left(\theta \right)}\right)}{r} \\ \dot{t} &= 1 \end{aligned} \]

Parameters

Symbol	Default	Role
tau	3	surface-actuation period

State variables: r, theta, t

Properties

Lyapunov spectrum

$+0.3539,\; -0.3556,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.995$

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.InteriorSquirmer()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Blake (1971), Bull. Aust. Math. Soc. 5, 255-264

BibTeX

@misc{interiorsquirmer,
  title = {InteriorSquirmer system},
  note = {Blake (1971), Bull. Aust. Math. Soc. 5, 255-264}
}