GumowskiMira¶

Systems / Maps / Exotic maps

The Gumowski–Mira map — accelerator-physics origin, endlessly varied ornamental attractors.

discrete · map2 dimensionsattractor

Definition¶

\[ \begin{aligned} y_{0}' &= a y_{0} + b y_{1} + \frac{y_{0}^{2} \left(2 - 2 a\right)}{y_{0}^{2} + 1} \\ y_{1}' &= a \left(a y_{0} + b y_{1} + \frac{y_{0}^{2} \left(2 - 2 a\right)}{y_{0}^{2} + 1}\right) - y_{0} + \frac{\left(2 - 2 a\right) \left(a y_{0} + b y_{1} + \frac{y_{0}^{2} \left(2 - 2 a\right)}{y_{0}^{2} + 1}\right)^{2}}{\left(a y_{0} + b y_{1} + \frac{y_{0}^{2} \left(2 - 2 a\right)}{y_{0}^{2} + 1}\right)^{2} + 1} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`-1.1`	shape coefficient of the rational nonlinearity G
`b`	`-0.2`	linear feedback gain mixing the previous y

Properties¶

Lyapunov spectrum

$-0.1585,\; -1.451$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 0$

Divergence ∇·f

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

Equilibria

1 fixed points
0 stable · 1 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Gumowski & Mira (1980), Recurrences and Discrete Dynamic Systems

BibTeX

@misc{gumowskimira,
  title = {GumowskiMira system},
  note = {Gumowski & Mira (1980), Recurrences and Discrete Dynamic Systems}
}