FoldedTowel

Systems / Maps / Chaotic maps

Rössler's folded-towel map — the prototypical hyperchaotic map, the lowest dimension admitting two positive Lyapunov exponents.

discrete · map3 dimensionshyperchaotic

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Definition

\[ \begin{aligned} x' &= a x \left(1 - x\right) - b \left(1 - 2 z\right) \left(c + y\right) \\ y' &= d \left(- e x + 1\right) \left(\left(1 - 2 z\right) \left(c + y\right) - 1\right) \\ z' &= f z \left(1 - z\right) + g y \end{aligned} \]

Parameters

Symbol	Default	Role
a	3.8	map parameter of the coupled quadratic recurrences
b	0.05	map parameter of the coupled quadratic recurrences
c	0.35	map parameter of the coupled quadratic recurrences
d	0.1	map parameter of the coupled quadratic recurrences
e	1.9	map parameter of the coupled quadratic recurrences
f	3.78	map parameter of the coupled quadratic recurrences
g	0.2	map parameter of the coupled quadratic recurrences

State variables: x, y, z

Properties

Lyapunov spectrum

$+0.43,\; +0.38,\; -3.3$
hyperchaotic map: two positive exponents; the widely quoted folded-towel spectrum is approximately (0.43, 0.37, -3.30), giving a Kaplan-Yorke dimension of about 2.25

Kaplan–Yorke dimension

$D_{KY} = 2.245$

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

5 fixed points
0 stable · 5 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.FoldedTowel()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Rössler (1979), 'Chaotic oscillations: an example of hyperchaos', Lectures in Applied Mathematics 17, 141-156

BibTeX

@misc{foldedtowel,
  title = {FoldedTowel system},
  note = {Rössler (1979), 'Chaotic oscillations: an example of hyperchaos', Lectures in Applied Mathematics 17, 141-156}
}