WindmiReduced

Systems / ODEs / Physical systems

A reduced model of the magnetosphere–ionosphere coupling (WINDMI) — substorm dynamics.

continuous · ODE3 dimensionschaotic

phase portrait

Definition

\[ \begin{aligned} \dot{i} &= a_{1} \left(- v + vsw\right) \\ \dot{v} &= b_{1} i - b_{2} \left|{p}\right|^{0.5} - b_{3} v \\ \dot{p} &= - \frac{vsw^{0.5} \left(1 - \tanh{\left(\frac{\left|{d_{1} \left(i - 1\right) - 25.0}\right|}{2} - \frac{\left|{d_{1} \left(i - 1\right) + 25.0}\right|}{2} \right)}\right) \left|{p}\right|^{1.25}}{2} + vsw^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a1	0.247	linear coupling coefficient of the reduced model
b1	10.8	linear coupling coefficient of the reduced model
b2	0.0752	linear coupling coefficient of the reduced model
b3	1.06	linear coupling coefficient of the reduced model
d1	2200	steepness of the tanh substorm-onset switch
vsw	5	solar-wind driver (input) strength

State variables: i, v, p

Properties

Lyapunov spectrum

$-3.935e-11,\; -0.5304,\; -0.5296$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 0$

Divergence ∇·f

n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Smith, Thiffeault & Horton (2000), J. Geophys. Res. 105, 12983-12996

BibTeX

@misc{windmireduced,
  title = {WindmiReduced system},
  note = {Smith, Thiffeault & Horton (2000), J. Geophys. Res. 105, 12983-12996}
}