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

\[ \begin{aligned} y_{0}' &= a y_{0} \left(1 - y_{0}\right) - b \left(1 - 2 y_{2}\right) \left(c + y_{1}\right) \\ y_{1}' &= d \left(- e y_{0} + 1\right) \left(\left(c + y_{1}\right) \left(2 y_{2} + 1\right) - 1\right) \\ y_{2}' &= f y_{2} \left(1 - y_{2}\right) + g y_{1} \end{aligned} \]

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

Properties

Lyapunov spectrum

≥ 2 positive exponents (literature — hyperchaotic map: two positive exponents)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

4 fixed points
0 stable · 4 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.FoldedTowel()
traj = sys.iterate(steps=10_000)

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

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}
}