Tinkerbell map

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

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

x_{n + 1} = x_{n}^{2} - y_{n}^{2} + a x_{n} + b y_{n}

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

y_{n + 1} = 2 x_{n} y_{n} + c x_{n} + d y_{n}

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

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

$a=0.9,b=-0.6013,c=2.0,d=0.50$
$a=0.3,b=0.6000,c=2.0,d=0.27$

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.

References[edit]

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[edit]

Tinkerbell map visualization with interactive source code

Chaos theory

Concepts

Core	Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space
Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy
Theorems	Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem

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Discrete	Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map
Continuous	Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator

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Tinkerbell map

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