LorenzStenflo

Systems / ODEs / Exotic systems

Stenflo's four-dimensional generalization of Lorenz for finite-amplitude acoustic-gravity waves in a rotating atmosphere — hyperchaotic for suitable parameters.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - a x + a y + d w \\ \dot{y} &= c x - x z - y \\ \dot{z} &= - b z + x y \\ \dot{w} &= - a w - x \end{aligned} \]

Parameters

Symbol	Default	Role
a	2	Prandtl-like number coupling x and y (and damping w)
b	0.7	geometric damping of the z mode
c	26	Rayleigh-like forcing in the y equation
d	1.5	rotation parameter coupling the auxiliary state w into x

State variables: x, y, z, w

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - 2 a - b - 1$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.LorenzStenflo()
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

Stenflo (1996), Phys. Scr. 53, 83-84
doi:10.1088/0031-8949/53/1/015

BibTeX

@misc{lorenzstenflo,
  title = {LorenzStenflo system},
  note = {Stenflo (1996), Phys. Scr. 53, 83-84},
  doi = {10.1088/0031-8949/53/1/015}
}