Qi

Systems / ODEs / Exotic systems

A 4-D flow in which every equation carries a triple cross-product term, giving a four-wing chaotic attractor far more complex than the quadratic Lorenz family.

continuous · ODE4 dimensionschaotic

projection (x, y, z)

projection (y, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	45	coupling rate of the x–y subsystem
b	10	linear gain in the y equation
c	1	damping of the z mode
d	10	damping of the w mode

Properties

Lyapunov spectrum

$+2.105,\; -0.00334,\; -4.394,\; -43.71$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.478$

Divergence ∇·f

$\nabla\!\cdot f = - a + b - c - d$
constant

Equilibria

9 equilibria
0 stable · 9 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Qi, van Wyk, van Wyk & Chen (2008), Phys. Lett. A 372, 124

BibTeX

@misc{qi,
  title = {Qi system},
  note = {Qi, van Wyk, van Wyk & Chen (2008), Phys. Lett. A 372, 124}
}