Tinkerbell¶

Systems / Maps / Chaotic maps

An ornamental 2-D quadratic map whose attractor resembles a fairy in flight.

discrete · map2 dimensionschaotic

Definition¶

\[ \begin{aligned} x' &= a x + b y + x^{2} - y^{2} \\ y' &= c x + d y + 2 x y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`0.9`	linear coefficient on x
`b`	`-0.6013`	linear coefficient on y
`c`	`2`	cross-coupling coefficient
`d`	`0.5`	cross-coupling coefficient

State variables: x, y

Properties¶

Lyapunov spectrum

≥ 1 positive exponent (literature — chaotic at default parameters)

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

2 fixed points
0 stable · 2 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Nusse & Yorke (1994), Dynamics: Numerical Explorations
doi:10.1007/978-1-4684-0231-5

BibTeX

@misc{tinkerbell,
  title = {Tinkerbell system},
  note = {Nusse & Yorke (1994), Dynamics: Numerical Explorations},
  doi = {10.1007/978-1-4684-0231-5}
}