HyperQi

Systems / ODEs / Exotic systems

A 4-D smooth quadratic system with two large positive Lyapunov exponents — a much-studied benchmark for hyperchaos synchronization and secure communication.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (y, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	50	coupling rate of the x–y subsystem
b	24	linear gain in the y equation
c	13	damping of the z mode
d	8	damping of the w mode
e	33	cross-coupling gain between the z and w states
f	30	cross-coupling gain between the z and w states

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

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

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.HyperQi()
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

Qi, van Wyk, van Wyk & Chen (2008), Phys. Lett. A 372, 124
doi:10.1016/j.physleta.2007.10.082

BibTeX

@misc{hyperqi,
  title = {HyperQi system},
  note = {Qi, van Wyk, van Wyk & Chen (2008), Phys. Lett. A 372, 124},
  doi = {10.1016/j.physleta.2007.10.082}
}