Kaplan–Yorke conjecture explained

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\geqλ} _{2\geq...\geqλ} _{n} , let j be the largest index for which\sum ^{j} _{i =1} λ _{i} \geqslant 0

and

\sum ^{j +1} _{i =1} λ _{i} < 0 .

Then the conjecture is that the dimension of the attractor is

D = j + (\sum ^{j λ} _{i})/(| λ _{j+1} |) .

This idea is used for the definition of the Lyapunov dimension.^[3]

Examples

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} and λ _{2=-2.34} . In this case, we find j = 1 and the dimension formula reduces toD = j + (λ _{1})/(=1+ | λ _{2|}) (0 . 603)/(|{ -2 . 34) |} =1 . 26 .

• The Lorenz system shows chaotic behavior at the parameter values

\sigma =16 , \rho =45 . 92 and \beta =4 . 0 . The resulting Lyapunov exponents are . Noting that j = 2, we findD =2+ (2 . 16+0 . 00)/(| -32 . 4 |) =2 . 07 .

Notes and References

1. Book: J. . Kaplan . J. . Yorke . James A. Yorke . Chaotic behavior of multidimensional difference equations . Functional Differential Equations and the Approximation of Fixed Points . Lecture Notes in Mathematics . 730 . H. O. . Peitgen . H. O. . Walther . Springer . Berlin . 1979 . 204–227 . 978-0-387-09518-9 . http://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1979_C11_Kaplan_multidimensional.pdf . 0547989.
2. P. . Frederickson . J. . Kaplan . E. . Yorke . J. . Yorke . The Lyapunov Dimension of Strange Attractors . . 49 . 1983 . 2 . 185–207 . 10.1016/0022-0396(83)90011-6 . 1983JDE....49..185F . free .
3. Book: Nikolay . Kuznetsov . Volker . Reitmann . 2020. Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Springer. Cham.
4. A. . Wolf . A. . Swift . B. . Jack . H. L. . Swinney . J. A. . Vastano . Determining Lyapunov Exponents from a Time Series . . 1985 . 16 . 3 . 285–317 . 10.1016/0167-2789(85)90011-9 . 1985PhyD...16..285W . 10.1.1.152.3162 . 14411384 .