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In applied mathematics, the Kaplan–Yorke conjecture concerns the
dimension of an
attractor , using
Lyapunov exponents .
[1]
[2] By arranging the Lyapunov exponents in order from largest to smallest
λ
1
≥
λ
2
≥
⋯
≥
λ
n
{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}}
, let j be the largest index for which
∑
i
=
1
j
λ
i
⩾
0
{\displaystyle \sum _{i=1}^{j}\lambda _{i}\geqslant 0}
and
∑
i
=
1
j
+
1
λ
i
<
0.
{\displaystyle \sum _{i=1}^{j+1}\lambda _{i}<0.}
Then the conjecture is that the dimension of the attractor is
D
=
j
+
∑
i
=
1
j
λ
i
|
λ
j
+
1
|
.
{\displaystyle D=j+{\frac {\sum _{i=1}^{j}\lambda _{i}}{|\lambda _{j+1}|}}.}
This idea is used for the definition of the
Lyapunov dimension .
[3]
Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to estimate the
fractal dimension
and the
Hausdorff dimension of the corresponding attractor.
[4]
[3]
The
Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents
λ
1
=
0.603
{\displaystyle \lambda _{1}=0.603}
and
λ
2
=
−
2.34
{\displaystyle \lambda _{2}=-2.34}
. In this case, we find j = 1 and the dimension formula reduces to
D
=
j
+
λ
1
|
λ
2
|
=
1
+
0.603
|
−
2.34
|
=
1.26.
{\displaystyle D=j+{\frac {\lambda _{1}}{|\lambda _{2}|}}=1+{\frac {0.603}{|{-2.34}|}}=1.26.}
The
Lorenz system shows chaotic behavior at the parameter values
σ
=
16
{\displaystyle \sigma =16}
,
ρ
=
45.92
{\displaystyle \rho =45.92}
and
β
=
4.0
{\displaystyle \beta =4.0}
. The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find
D
=
2
+
2.16
+
0.00
|
−
32.4
|
=
2.07.
{\displaystyle D=2+{\frac {2.16+0.00}{|-32.4|}}=2.07.}
^ Kaplan, J.;
Yorke, J. (1979).
"Chaotic behavior of multidimensional difference equations" (PDF) . In Peitgen, H. O.; Walther, H. O. (eds.). Functional Differential Equations and the Approximation of Fixed Points . Lecture Notes in Mathematics. Vol. 730. Berlin: Springer. pp. 204–227.
ISBN
978-0-387-09518-9 .
MR
0547989 .
^ Frederickson, P.; Kaplan, J.; Yorke, E.; Yorke, J. (1983).
"The Lyapunov Dimension of Strange Attractors" .
J. Diff. Eqs. 49 (2): 185–207.
Bibcode :
1983JDE....49..185F .
doi :
10.1016/0022-0396(83)90011-6 .
^
a
b Kuznetsov, Nikolay; Reitmann, Volker (2020).
Attractor Dimension Estimates for Dynamical Systems: Theory and Computation . Cham: Springer.
^ Wolf, A.; Swift, A.; Jack, B.; Swinney, H. L.; Vastano, J. A. (1985). "Determining Lyapunov Exponents from a Time Series".
Physica D . 16 (3): 285–317.
Bibcode :
1985PhyD...16..285W .
CiteSeerX
10.1.1.152.3162 .
doi :
10.1016/0167-2789(85)90011-9 .
S2CID
14411384 .