AtmosphericRegime¶

Systems / ODEs / Climate & geophysics

Two coupled oscillators with widely spaced frequencies whose timescale separation drives intermittent bursts between quasi-stationary weather regimes.

continuous · ODE3 dimensionsintermittent

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= \mu_{1} x + \sigma x y \\ \dot{y} &= \alpha y z + \beta z^{2} + \mu_{2} y + \omega z - \sigma x^{2} \\ \dot{z} &= - \alpha y^{2} - \beta y z + \mu_{2} z - \omega y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`alpha`	`-2`	quadratic slow–fast coupling coefficient
`beta`	`-5`	quadratic slow–fast coupling coefficient
`mu1`	`0.05`	linear growth/damping of the slow mode
`mu2`	`-0.01`	linear growth/damping of the fast oscillator
`omega`	`3`	natural frequency of the fast oscillator
`sigma`	`1.1`	bilinear slow–fast feedback strength

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = \alpha z - \beta y + \mu_{1} + 2 \mu_{2} + \sigma y$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.AtmosphericRegime()
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¶

Tuwankotta (2006), Int. J. Non-Linear Mech. 41, 180-191
doi:10.1016/j.ijnonlinmec.2005.02.007

BibTeX

@misc{atmosphericregime,
  title = {AtmosphericRegime system},
  note = {Tuwankotta (2006), Int. J. Non-Linear Mech. 41, 180-191},
  doi = {10.1016/j.ijnonlinmec.2005.02.007}
}