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Kaplan–Yorke conjecture

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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 , let j be the largest index for which

and

Then the conjecture is that the dimension of the attractor is

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

Examples

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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 and . In this case, we find j = 1 and the dimension formula reduces to
  • The Lorenz system shows chaotic behavior at the parameter values , and . The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find

References

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  1. ^ 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.
  2. ^ 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.
  3. ^ a b Kuznetsov, Nikolay; Reitmann, Volker (2020). Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Cham: Springer.
  4. ^ 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.