Editing Lyapunov exponent (section)

==Significance of the Lyapunov spectrum==

The Lyapunov spectrum can be used to give an estimate of the rate of entropy production,
of the [[fractal dimension]], and of the [[Hausdorff dimension]] of the considered [[dynamical system]].<ref name=2020-KuznetsovR>{{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}}</ref> In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called [[Lyapunov dimension]] (or [[Kaplan–Yorke conjecture|Kaplan–Yorke dimension]]) <math> D_{KY} </math>, which is defined as follows:
<math display="block"> D_{KY}= k + \sum_{i=1}^k \frac{\lambda_i}{|\lambda_{k+1}|} </math>
where <math> k </math> is the maximum integer such that the sum of the <math> k </math> largest exponents is still non-negative. <math> D_{KY} </math> represents an upper bound for the [[information dimension]] of the system.<ref name=ky>{{Cite book |first1=J. |last1=Kaplan |name-list-style=amp |first2=J. |last2=Yorke |chapter=Chaotic behavior of multidimensional difference equations |editor1-last=Peitgen |editor1-first=H. O. |editor2-last=Walther |editor2-first=H. O. |title=Functional Differential Equations and Approximation of Fixed Points |publisher=Springer |location=New York |year=1979 |isbn=978-3-540-09518-7 }}</ref> Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the [[Kolmogorov–Sinai entropy]] accordingly to Pesin's theorem.<ref name=pesin>{{Cite journal |first=Y. B. |last=Pesin |title=Characteristic Lyapunov Exponents and Smooth Ergodic Theory |journal=Russian Math. Surveys |volume=32 |year=1977 |issue=4 |pages=55–114 |doi= 10.1070/RM1977v032n04ABEH001639|bibcode = 1977RuMaS..32...55P |s2cid=250877457 }}</ref>
Along with widely used numerical methods for estimating and computing the [[Lyapunov dimension]] there is an effective analytical approach, which is based on the direct Lyapunov method with special Lyapunov-like functions.<ref>{{Cite journal |first=N.V. |last=Kuznetsov |title=The Lyapunov dimension and its estimation via the Leonov method |journal=Physics Letters A |volume=380 |year=2016 |issue=25–26 |pages=2142–2149 |doi= 10.1016/j.physleta.2016.04.036|bibcode =2016PhLA..380.2142K |arxiv=1602.05410|s2cid=118467839 }}</ref>
The Lyapunov exponents of bounded trajectory and the [[Lyapunov dimension]] of attractor are invariant under [[diffeomorphism]] of the phase space.<ref>{{Cite journal
|first1=N.V. |last1=Kuznetsov
|first2=T.A. |last2=Alexeeva
|first3=G.A. |last3=Leonov
|title=Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations
|journal=Nonlinear Dynamics |volume=85 |year=2016 |issue=1 |pages=195–201 |doi= 10.1007/s11071-016-2678-4|arxiv=1410.2016|bibcode=2016NonDy..85..195K
|s2cid=119650438
}}</ref>

The [[multiplicative inverse]] of the largest Lyapunov exponent is sometimes referred in literature as [[Lyapunov time]], and defines the characteristic ''e''-folding time. For chaotic orbits, the Lyapunov time will be finite, whereas for regular orbits it will be infinite.