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A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x 0 ,y0 ) was (128873/350377,0.667751).
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
=
2
x
n
(
mod
1
)
{\displaystyle x_{n+1}=2x_{n}\ ({\textrm {mod}}~1)}
y
n
+
1
=
α
y
n
+
cos
(
4
π
x
n
)
{\displaystyle 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 [ edit ]
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
=
2
a
n
(
mod
b
)
{\displaystyle a_{n+1}=2a_{n}\ ({\textrm {mod}}~b)}
x
n
+
1
=
a
n
/
b
{\displaystyle x_{n+1}=a_{n}/b}
y
n
+
1
=
α
y
n
+
cos
(
4
π
x
n
)
{\displaystyle y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})}
where the
a
n
{\displaystyle a_{n}}
and
b
{\displaystyle b}
are computational integers. It is also best to choose
b
{\displaystyle b}
to be a large prime number in order to get many different values of
x
n
{\displaystyle x_{n}}
.
Another way to avoid having the modulo operator yield zero after a short number of iterations is
x
n
+
1
=
2
x
n
(
mod
0.99995
)
{\displaystyle x_{n+1}=2x_{n}\ ({\textrm {mod}}~0.99995)}
y
n
+
1
=
α
y
n
+
cos
(
4
π
x
n
)
{\displaystyle y_{n+1}=\alpha y_{n}+\cos(4\pi x_{n})}
which will still eventually return zero, albeit after many more iterations.
References [ edit ]
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 .
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R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Kaplan–Yorke_map&oldid=1099211391 "
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