LidDrivenCavityFlow¶
Systems / ODEs / Climate & geophysics
A time-periodic lid-driven cavity whose two-mode wall driving reverses each period, braiding passive tracers into topological chaos.
Definition¶
\[
\begin{aligned}
\dot{x} &= \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(- \frac{2 u_{1} \left(- \pi b \sinh{\left(\frac{\pi b}{a} \right)} \sinh{\left(\frac{\pi y}{a} \right)} + \left(a \sinh{\left(\frac{\pi y}{a} \right)} + \pi y \cosh{\left(\frac{\pi y}{a} \right)}\right) \cosh{\left(\frac{\pi b}{a} \right)}\right) \sin{\left(\frac{\pi x}{a} \right)}}{a \sinh{\left(\frac{2 \pi b}{a} \right)} + 2 \pi b} + \frac{2 u_{2} \left(- 2 \pi b \sinh{\left(\frac{2 \pi b}{a} \right)} \sinh{\left(\frac{2 \pi y}{a} \right)} + \left(a \sinh{\left(\frac{2 \pi y}{a} \right)} + 2 \pi y \cosh{\left(\frac{2 \pi y}{a} \right)}\right) \cosh{\left(\frac{2 \pi b}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{a} \right)}}{a \sinh{\left(\frac{4 \pi b}{a} \right)} + 4 \pi b}\right) + \left(\frac{2 u_{1} \left(- \pi b \sinh{\left(\frac{\pi b}{a} \right)} \sinh{\left(\frac{\pi y}{a} \right)} + \left(a \sinh{\left(\frac{\pi y}{a} \right)} + \pi y \cosh{\left(\frac{\pi y}{a} \right)}\right) \cosh{\left(\frac{\pi b}{a} \right)}\right) \sin{\left(\frac{\pi x}{a} \right)}}{a \sinh{\left(\frac{2 \pi b}{a} \right)} + 2 \pi b} + \frac{2 u_{2} \left(- 2 \pi b \sinh{\left(\frac{2 \pi b}{a} \right)} \sinh{\left(\frac{2 \pi y}{a} \right)} + \left(a \sinh{\left(\frac{2 \pi y}{a} \right)} + 2 \pi y \cosh{\left(\frac{2 \pi y}{a} \right)}\right) \cosh{\left(\frac{2 \pi b}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{a} \right)}}{a \sinh{\left(\frac{4 \pi b}{a} \right)} + 4 \pi b}\right) \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \\
\dot{y} &= \left(0.5 - 0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)}\right) \left(- \frac{2 \pi u_{1} \left(b \sinh{\left(\frac{\pi b}{a} \right)} \cosh{\left(\frac{\pi y}{a} \right)} - y \sinh{\left(\frac{\pi y}{a} \right)} \cosh{\left(\frac{\pi b}{a} \right)}\right) \cos{\left(\frac{\pi x}{a} \right)}}{a \sinh{\left(\frac{2 \pi b}{a} \right)} + 2 \pi b} + \frac{4 \pi u_{2} \left(b \sinh{\left(\frac{2 \pi b}{a} \right)} \cosh{\left(\frac{2 \pi y}{a} \right)} - y \sinh{\left(\frac{2 \pi y}{a} \right)} \cosh{\left(\frac{2 \pi b}{a} \right)}\right) \cos{\left(\frac{2 \pi x}{a} \right)}}{a \sinh{\left(\frac{4 \pi b}{a} \right)} + 4 \pi b}\right) + \left(\frac{2 \pi u_{1} \left(b \sinh{\left(\frac{\pi b}{a} \right)} \cosh{\left(\frac{\pi y}{a} \right)} - y \sinh{\left(\frac{\pi y}{a} \right)} \cosh{\left(\frac{\pi b}{a} \right)}\right) \cos{\left(\frac{\pi x}{a} \right)}}{a \sinh{\left(\frac{2 \pi b}{a} \right)} + 2 \pi b} + \frac{4 \pi u_{2} \left(b \sinh{\left(\frac{2 \pi b}{a} \right)} \cosh{\left(\frac{2 \pi y}{a} \right)} - y \sinh{\left(\frac{2 \pi y}{a} \right)} \cosh{\left(\frac{2 \pi b}{a} \right)}\right) \cos{\left(\frac{2 \pi x}{a} \right)}}{a \sinh{\left(\frac{4 \pi b}{a} \right)} + 4 \pi b}\right) \left(0.5 \tanh{\left(20 \tau \sin{\left(\frac{2 \pi t}{\tau} \right)} \right)} + 0.5\right) \\
\dot{t} &= 1
\end{aligned}
\]
Parameters¶
| Symbol | Default | Role |
|---|---|---|
a |
6 |
cavity width |
b |
1 |
cavity half-height |
tau |
1.1 |
wall-reversal period |
u1 |
9.92786 |
first driving-mode amplitude |
u2 |
8.34932 |
second driving-mode amplitude |
State variables: x, y, t
Properties¶
Lyapunov spectrum
TODO — lyapunov_spectrum failed: AttributeError
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
$\nabla\!\cdot f = 0$
constant
constant
Equilibria
none found (no equilibria)
Use it in TSDynamics¶
import tsdynamics as ts
system = ts.systems.LidDrivenCavityFlow()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)
spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)
Reference¶
Grover, Ross, Stremler & Kumar (2012), Chaos 22, 043135
doi:10.1063/1.4768666