Editing Lyapunov exponent (section)

==Numerical calculation==
[[File:Lyapunov exponents of the Mandelbrot set (The mini-Mandelbrot) - Matlab.png|alt=Lyapunov exponent|left|thumb|220x220px|Points inside and outside [[Mandelbrot set]] colored by Lyapunov exponent.]]
Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by [[Richard H. Miller|R. H. Miller]] in 1964.<ref>{{Cite journal | doi = 10.1086/147911| title = Irreversibility in Small Stellar Dynamical Systems| journal = The Astrophysical Journal| volume = 140| pages = 250| year = 1964| last1 = Miller | first1 = R. H.|bibcode = 1964ApJ...140..250M }}</ref> Currently, the most commonly used numerical procedure estimates the <math>L</math> matrix based on averaging several finite time approximations of the limit defining <math>L</math>.

One of the most used and effective numerical techniques to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic [[Gram–Schmidt]] orthonormalization of the [[Lyapunov vector]]s to avoid a misalignment of all the vectors along the direction of maximal expansion.<ref name=benettin>{{Cite journal | last1 = Benettin | first1 = G. | last2 = Galgani | first2 = L. | last3 = Giorgilli | first3 = A. | last4 = Strelcyn | first4 = J. M. | title = Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory | doi = 10.1007/BF02128236 | journal = Meccanica | volume = 15 | pages = 9–20 | year = 1980 | s2cid = 123085922 }}</ref><ref>{{Cite journal | last1 = Benettin | first1 = G. | last2 = Galgani | first2 = L. | last3 = Giorgilli | first3 = A. | last4 = Strelcyn | first4 = J. M. | title = Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application | doi = 10.1007/BF02128237 | journal = Meccanica | volume = 15 | pages = 21–30 | year = 1980 | s2cid = 117095512 }}</ref><ref name=shimada>{{Cite journal | last1 = Shimada | first1 = I. | last2 = Nagashima | first2 = T. | doi = 10.1143/PTP.61.1605 | title = A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems | journal = Progress of Theoretical Physics | volume = 61 | issue = 6 | pages = 1605–1616 | year = 1979 |bibcode = 1979PThPh..61.1605S | doi-access = free }}</ref><ref>{{Cite journal | year =  1985| title = Ergodic theory of chaos and strange attractors | journal =  Reviews of Modern Physics| volume =  57| issue =  3| pages =  617–656| doi = 10.1103/RevModPhys.57.617 |bibcode = 1985RvMP...57..617E | last1 = Eckmann | first1 = J. -P. | last2 = Ruelle | first2 = D. | s2cid = 18330392 }}</ref> The Lyapunov spectrum of various models are described.<ref>{{Cite book |last=Sprott |first=Julien Clinton |title=Chaos and Time-Series Analysis |publisher=Oxford University Press |date=September 27, 2001 |isbn=978-0198508403}}</ref> Source codes for nonlinear systems such as the Hénon map, the Lorenz equations, a delay differential equation and so on are introduced.<ref>{{Cite web |last=Sprott |first=Julien Clinton |date=May 26, 2005 |title=Lyapunov Exponent Spectrum Software |url=https://sprott.physics.wisc.edu/chaos/lespec.htm}}</ref><ref>{{Cite web |last=Sprott |first=Julien Clinton |date=October 4, 2006 |title=Lyapunov Exponents for Delay Differential Equations |url=https://sprott.physics.wisc.edu/chaos/ddele.htm}}</ref><ref>{{Cite web |last=Tomo |first=Nakamura |date=19 October 2022 |title=nonlinear systems and Lyapunov spectrum |url=https://sites.google.com/view/lyapunov-spectrum/home |archive-date=}}</ref>

For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed. However, there are many difficulties with applying these methods and such problems should be approached with care.  The main difficulty is that the data does not fully explore the phase space, rather it is confined to the attractor which has very limited (if any) extension along certain directions.  These thinner or more singular directions within the data set are the ones associated with the more negative exponents.  The use of nonlinear mappings to model the evolution of small displacements from the attractor has been shown to dramatically improve the ability to recover the Lyapunov spectrum,<ref name=Bryant1>{{Cite journal | doi = 10.1103/PhysRevLett.65.1523| pmid = 10042292| title = Lyapunov exponents from observed time series| journal = Physical Review Letters| volume = 65| issue = 13| pages = 1523–1526| year = 1990| last1 = Bryant | first1 = P. | last2 = Brown | first2 = R. | last3 = Abarbanel | first3 = H. |bibcode = 1990PhRvL..65.1523B }}</ref><ref name=brown>{{Cite journal | doi = 10.1103/PhysRevA.43.2787| pmid = 9905344| title = Computing the Lyapunov spectrum of a dynamical system from an observed time series| journal = Physical Review A| volume = 43| issue = 6| pages = 2787–2806| year = 1991| last1 = Brown | first1 = R. | last2 = Bryant | first2 = P. | last3 = Abarbanel | first3 = H. |bibcode = 1991PhRvA..43.2787B }}</ref> provided the data has a very low level of noise.  The singular nature of the data and its connection to the more negative exponents has also been explored.<ref name=bryant2>{{Cite journal | doi = 10.1016/0375-9601(93)91136-S| title = Extensional singularity dimensions for strange attractors| journal = Physics Letters A| volume = 179| issue = 3| pages = 186–190| year = 1993| last1 = Bryant | first1 = P. H. |bibcode = 1993PhLA..179..186B }}</ref>