Kaplan–Yorke map

The Kaplan–Yorke map is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior. The Kaplan–Yorke map takes a point (x_n, y_n ) in the plane and maps it to a new point given by

x_{n+1}=2x_{n}\ ({\textrm {mod}}~1)

y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})

where mod is the modulo operator with real arguments. The map depends on only the one constant α.

Calculation method

Due to roundoff error, successive applications of the modulo operator will yield zero after some ten or twenty iterations when implemented as a floating point operation on a computer. It is better to implement the following equivalent algorithm:

a_{n+1}=2a_{n}\ ({\textrm {mod}}~b)

x_{n+1}=a_{n}/b

y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})

where the $a_{n}$ and $b$ are computational integers. It is also best to choose $b$ to be a large prime number in order to get many different values of $x_{n}$ .

Another way to avoid having the modulo operator yield zero after a short number of iterations is

x_{n+1}=2x_{n}\ ({\textrm {mod}}~0.99995)

y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})

which will still eventually return zero, albeit after many more iterations.

References

J.L. Kaplan and J.A. Yorke (1979). H.O. Peitgen and H.O. Walther (ed.). Functional Differential Equations and Approximations of Fixed Points (Lecture Notes in Mathematics 730). Springer-Verlag. ISBN 0-387-09518-7.
P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica. 9D (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.

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

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