GumowskiMira¶

Systems / Maps / Exotic maps

The Gumowski–Mira map — accelerator-physics origin, endlessly varied ornamental phase portraits. At the shipped defaults it is area-preserving: a chaotic sea threaded with periodic islands rather than an attractor.

discrete · map2 dimensionsconservativechaotic

Definition¶

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

Parameters¶

Symbol	Default	Role
`a`	`-0.35`	shape coefficient of the rational nonlinearity G
`b`	`1`	linear feedback gain mixing the previous y

State variables: x, y

Properties¶

Lyapunov spectrum

≥ 1 positive exponent (literature — measured (QR tangent-map spectrum): lambda_1 ~ +0.13 from the chaotic sea, stable over 5*10**6 iterations, with lambda_2 = -lambda_1 since det J = b = 1. Only the sign is pinned: Hamiltonian stickiness makes the finite-time value window-dependent (+0.10 to +0.15))

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

3 fixed points
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Gumowski & Mira (1980), Recurrences and Discrete Dynamic Systems
doi:10.1007/bfb0089135

BibTeX

@misc{gumowskimira,
  title = {GumowskiMira system},
  note = {Gumowski & Mira (1980), Recurrences and Discrete Dynamic Systems},
  doi = {10.1007/bfb0089135}
}