Gingerbreadman map

Chaotic map

Gingerbreadman map for subset

Q^{2},[-10..10,-10..10]

{\displaystyle Q^{2},[-10..10,-10..10]}: the color of each point is related to the relative orbit period. To view the gingerbread man, you must rotate the image 135 degrees clockwise.

In dynamical systems theory, the Gingerbreadman map is a chaotic two-dimensional map. It is given by the piecewise linear transformation:^[1]

{\begin{cases}x_{n+1}=1-y_{n}+|x_{n}|\\y_{n+1}=x_{n}\end{cases}}

{\displaystyle {\begin{cases}x_{n+1}=1-y_{n}+|x_{n}|\\y_{n+1}=x_{n}\end{cases}}}

A crude Gingerbreadman map made using the turtle library in python.

References

[edit ]

^ Devaney, Robert L. (1988), "Fractal patterns arising in chaotic dynamical systems", in Peitgen, Heinz-Otto; Saupe, Dietmar (eds.), The Science of Fractal Images, Springer-Verlag, pp. 137–168, doi:10.1007/978-1-4612-3784-6_3 . See in particular Fig. 3.3.

External links

[edit ]

Weisstein, Eric W. "Gingerbreadman Map". MathWorld .

v
t
e

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

Conus textile shell

Circle map with black Arnold tongues

Theoretical
branches

Chaotic
maps (list)

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

Physical
systems

Chaos
theorists

See also

References

External links