HyperLorenz

Systems / ODEs / Chaotic attractors

A four-dimensional Lorenz extension with a feedback state that lifts the flow into hyperchaos — two positive Lyapunov exponents.

continuous · ODE4 dimensionshyperchaotic

projection (x, y, z)

projection (x, z, w)

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	10	Lorenz-core coefficient
b	2.667	Lorenz-core coefficient
c	28	Lorenz-core coefficient
d	1.1	fourth-variable feedback coefficient

Properties

Lyapunov spectrum

$+0.3192,\; +0.03068,\; +0.003099,\; -12.92$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3.027$

Divergence ∇·f

$\nabla\!\cdot f = - a - b + d - 1$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Meier (2003), Presentation of Attractors with Cinema

BibTeX

@misc{hyperlorenz,
  title = {HyperLorenz system},
  note = {Meier (2003), Presentation of Attractors with Cinema}
}