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

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

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

Properties

Lyapunov spectrum

$+12.45,\; +2.389,\; -0.02659,\; -61.82$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.24$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.HyperQi()
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{hyperqi,
  title = {HyperQi system},
  note = {Qi, van Wyk, van Wyk & Chen (2008), Phys. Lett. A 372, 124}
}