Kaplan–Yorke conjecture

In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents.{{cite book |first1=J. |last1=Kaplan |first2=J. |last2=Yorke |author-link2=James A. Yorke |chapter=Chaotic behavior of multidimensional difference equations |title=Functional Differential Equations and the Approximation of Fixed Points |series=Lecture Notes in Mathematics |volume=730 |editor-first=H. O. |editor-last=Peitgen |editor2-first=H. O. |editor2-last=Walther |publisher=Springer |location=Berlin |year=1979 |pages=204–227 |isbn=978-0-387-09518-9 |chapter-url=http://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1979_C11_Kaplan_multidimensional.pdf |mr=0547989}}{{cite journal |first1=P. |last1=Frederickson |first2=J. |last2=Kaplan |first3=E. |last3=Yorke |first4=J. |last4=Yorke |title=The Lyapunov Dimension of Strange Attractors |journal=J. Diff. Eqs. |volume=49 |year=1983 |issue=2 |pages=185–207 |doi=10.1016/0022-0396(83)90011-6 |bibcode=1983JDE....49..185F |doi-access=free }} By arranging the Lyapunov exponents in order from largest to smallest $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$ , let j be the largest index for which

: $\sum_{i=1}^j \lambda_i \geqslant 0$

and

: $\sum_{i=1}^{j+1} \lambda_i < 0.$

Then the conjecture is that the dimension of the attractor is

: $D=j+\frac{\sum_{i=1}^j\lambda_i}$

\lambda_{j+1}

This idea is used for the definition of the Lyapunov dimension.{{cite book | first1= Nikolay | last1=Kuznetsov | first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation| publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}

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.{{cite journal |first1=A. |last1=Wolf |first2=A. |last2=Swift |first3=B. |last3=Jack |first4=H. L. |last4=Swinney |first5=J. A. |last5=Vastano |title=Determining Lyapunov Exponents from a Time Series |journal=Physica D |year=1985 |volume=16 |issue=3 |pages=285–317 |doi=10.1016/0167-2789(85)90011-9 |bibcode=1985PhyD...16..285W |citeseerx=10.1.1.152.3162 |s2cid=14411384 }}

The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents $\lambda_1=0.603$ and $\lambda_2=-2.34$ . In this case, we find j = 1 and the dimension formula reduces to

:: $D=j+\frac{\lambda_1}$

\lambda_2

=1+\frac{0.603}

{-2.34}

=1.26.

The Lorenz system shows chaotic behavior at the parameter values $\sigma=16$ , $\rho=45.92$ and $\beta=4.0$ . The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find

:: $D=2+\frac{2.16 + 0.00}$

-32.4

=2.07.

References

Category:Dimension

Category:Dynamical systems

Category:Limit sets

Category:Conjectures