HyperLu

Systems / ODEs / Exotic systems

A 4-D hyperchaotic flow from state-feedback control of the Lü system, with two positive Lyapunov exponents verified by an electronic-circuit realization.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	36	coupling rate of the x–y (Lü) subsystem
b	3	damping of the z mode
c	20	self-gain of the y mode
d	1.3	self-gain of the auxiliary feedback state w

Properties

Lyapunov spectrum

$+0.8957,\; +0.4701,\; +0.009383,\; -19.08$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.072$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Chen, Lu, Lü & Yu (2006), Physica A 364, 103

BibTeX

@misc{hyperlu,
  title = {HyperLu system},
  note = {Chen, Lu, Lü & Yu (2006), Physica A 364, 103}
}