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

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} + d y_{3} \\ \dot{y_{1}} &= c y_{0} - y_{0} y_{2} - y_{1} \\ \dot{y_{2}} &= - b y_{2} + y_{0} y_{1} \\ \dot{y_{3}} &= - a y_{3} - y_{0} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.3722,\; +0.004845,\; -2.63,\; -3.447$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.143$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Stenflo (1996), Phys. Scr. 53, 83-84

BibTeX

@misc{lorenzstenflo,
  title = {LorenzStenflo system},
  note = {Stenflo (1996), Phys. Scr. 53, 83-84}
}