Tinkerbell map

The Tinkerbell map is a discrete-time dynamical system given by:

Tinkerbell attractor with a=0.9, b=-0.6013, c=2, d=0.5. Used starting values of ${\displaystyle x_{0}=-0.72}$ and ${\displaystyle y_{0}=-0.64}$.

${\displaystyle x_{n+1}=x_{n}^{2}-y_{n}^{2}+ax_{n}+by_{n}}$

${\displaystyle y_{n+1}=2x_{n}y_{n}+cx_{n}+dy_{n}}$

Some commonly used values of a, b, c, and d are

• ${\displaystyle a=0.9,b=-0.6013,c=2.0,d=0.50}$
• ${\displaystyle a=0.3,b=0.6000,c=2.0,d=0.27}$

Like all chaotic maps, the Tinkerbell Map has also been shown to have periods; after a certain number of mapping iterations any given point shown in the map to the right will find itself once again at its starting ___location.

The origin of the name is uncertain; however, the graphical picture of the system (as shown to the right) shows a similarity to the movement of Tinker Bell over Cinderella Castle, as shown at the beginning of all films produced by Disney.

Tinkerbell attractor with a=0.9, b=-0.6013, c=2. Used starting values of Xo = -0.7, Yo= -0.6. I vary d value from 0.5 to 0.4.

See also

• List of chaotic maps

References

• C.L. Bremer & D.T. Kaplan, Markov Chain Monte Carlo Estimation of Nonlinear Dynamics from Time Series
• K.T. Alligood, T.D. Sauer & J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Berlin: Springer-Verlag, 1996.
• P.E. McSharry & P.R.C. Ruffino, Asymptotic angular stability in non-linear systems: rotation numbers and winding numbers
• R.L. Davidchack, Y.-C. Lai, A. Klebanoff & E.M. Bollt, Towards complete detection of unstable periodic orbits in chaotic systems
• B. R. Hunt, Judy A. Kennedy, Tien-Yien Li, Helena E. Nusse, "SLYRB measures: natural invariant measures for chaotic systems"
• A. Goldsztejn, W. Hayes, P. Collins "Tinkerbell is Chaotic" SIAM J. Applied Dynamical Systems 10, n.4 1480-1501, 2011

External links

• Tinkerbell map visualization with interactive source code