LorenzBounded

Systems / ODEs / Chaotic attractors

The Lorenz attractor reformulated with polynomial corrections that confine it to a bounded region, recovering the standard flow as r grows.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - \sigma x + \sigma y + \frac{\sigma x^{3}}{r^{2}} - \frac{\sigma x^{2} y}{r^{2}} + \frac{\sigma x y^{2}}{r^{2}} + \frac{\sigma x z^{2}}{r^{2}} - \frac{\sigma y^{3}}{r^{2}} - \frac{\sigma y z^{2}}{r^{2}} \\ \dot{y} &= \rho x - x z - y - \frac{\rho x^{3}}{r^{2}} - \frac{\rho x y^{2}}{r^{2}} - \frac{\rho x z^{2}}{r^{2}} + \frac{x^{3} z}{r^{2}} + \frac{x^{2} y}{r^{2}} + \frac{x y^{2} z}{r^{2}} + \frac{x z^{3}}{r^{2}} + \frac{y^{3}}{r^{2}} + \frac{y z^{2}}{r^{2}} \\ \dot{z} &= - \beta z + \frac{\beta x^{2} z}{r^{2}} + \frac{\beta y^{2} z}{r^{2}} + \frac{\beta z^{3}}{r^{2}} + x y - \frac{x^{3} y}{r^{2}} - \frac{x y^{3}}{r^{2}} - \frac{x y z^{2}}{r^{2}} \end{aligned} \]

Parameters

Symbol	Default	Role
beta	2.667	geometric aspect ratio
r	64	bounding radius (larger r recovers unbounded Lorenz)
rho	28	Rayleigh ratio
sigma	10	Prandtl number

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - \beta + \frac{\beta x^{2}}{r^{2}} + \frac{\beta y^{2}}{r^{2}} + \frac{3 \beta z^{2}}{r^{2}} - \sigma - 1 - \frac{2 \rho x y}{r^{2}} + \frac{3 \sigma x^{2}}{r^{2}} - \frac{2 \sigma x y}{r^{2}} + \frac{\sigma y^{2}}{r^{2}} + \frac{\sigma z^{2}}{r^{2}} + \frac{x^{2}}{r^{2}} + \frac{3 y^{2}}{r^{2}} + \frac{z^{2}}{r^{2}}$
state-dependent

Equilibria

154 equilibria
0 stable · 154 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.LorenzBounded()
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

Sprott & Xiong (2015), Chaos 25, 083101
doi:10.1063/1.4927643

BibTeX

@misc{lorenzbounded,
  title = {LorenzBounded system},
  note = {Sprott & Xiong (2015), Chaos 25, 083101},
  doi = {10.1063/1.4927643}
}