Editing Lyapunov exponent (section)

==Further reading==
* {{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}}
* {{cite journal
 |author1=M.-F. Danca |author2=N.V. Kuznetsov |name-list-style=amp | year = 2018
 | title = Matlab Code for Lyapunov Exponents of Fractional-Order Systems
 | journal = International Journal of Bifurcation and Chaos
 | volume = 25
 | pages = 1850067–1851392
 | doi = 10.1142/S0218127418500670
 | issue = 5
 | arxiv = 1804.01143
 |bibcode=2018IJBC...2850067D }}
* Cvitanović P., Artuso  R., Mainieri R., Tanner G.  and  Vattay G.[http://www.chaosbook.org/ Chaos: Classical and Quantum] Niels Bohr Institute, Copenhagen 2005 – ''textbook about chaos available under [[Free Documentation License]]''
* {{cite journal
 |author1=Freddy Christiansen
 |author2=Hans Henrik Rugh
 |name-list-style=amp
 |year=1997
 |title=Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization
 |journal=Nonlinearity
 |volume=10
 |issue=5
 |pages=1063–1072
 |url=http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
 |doi=10.1088/0951-7715/10/5/004
 |bibcode=1997Nonli..10.1063C
 |url-status=dead
 |archive-url=https://web.archive.org/web/20060425194442/http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
 |archive-date=2006-04-25
 | arxiv=chao-dyn/9611014
 |s2cid=122976405
 }}
* {{cite journal
 |author1=Salman Habib  |author2=Robert D. Ryne
  | name-list-style=amp | year = 1995
 | title = Symplectic Calculation of Lyapunov Exponents
 | journal = Physical Review Letters
 | volume = 74
 | issue = 1
 | pages = 70–73
 | arxiv = chao-dyn/9406010
 | doi = 10.1103/PhysRevLett.74.70
| pmid=10057701
| bibcode=1995PhRvL..74...70H|s2cid=19203665
 }}
* {{cite journal
 |author1=Govindan Rangarajan |author2=Salman Habib |author3=Robert D. Ryne  |name-list-style=amp | year = 1998
 | title = Lyapunov Exponents without Rescaling and Reorthogonalization
 | journal = Physical Review Letters
 | volume = 80
 | issue = 17
 | pages = 3747–3750
 | arxiv = chao-dyn/9803017
 | doi = 10.1103/PhysRevLett.80.3747
| bibcode=1998PhRvL..80.3747R|s2cid=14483592 }}
* {{cite journal
 |author1=X. Zeng |author2=R. Eykholt |author3=R. A. Pielke  |name-list-style=amp | year = 1991
 | title = Estimating the Lyapunov-exponent spectrum from short time series of low precision
 | journal = Physical Review Letters
 | volume = 66
 | pages = 3229–3232
 | doi = 10.1103/PhysRevLett.66.3229
 | pmid = 10043734
 | issue = 25
| bibcode=1991PhRvL..66.3229Z
}}
* {{cite journal
 |author1=E Aurell |author2=G Boffetta |author3=A Crisanti |author4=G Paladin |author5=A Vulpiani | year = 1997
 | title = Predictability in the large: an extension of the concept of Lyapunov exponent
 | volume = 30
 | issue = 1
 | pages = 1–26
 | journal = J. Phys. A: Math. Gen.
 | doi = 10.1088/0305-4470/30/1/003
 |bibcode = 1997JPhA...30....1A |arxiv=chao-dyn/9606014|s2cid=54697488 }}
* {{cite journal
 |author1=F Ginelli |author2=P Poggi |author3=A Turchi |author4=H Chaté |author5=R Livi |author6=A Politi | year = 2007
 | title =  Characterizing Dynamics with Covariant Lyapunov Vectors
 | volume = 99
 | pages =   130601
 | url = http://www.fi.isc.cnr.it/users/antonio.politi/Reprints/145.pdf
 | archive-url = https://wayback.archive-it.org/all/20081031185653/http://www.fi.isc.cnr.it/users/antonio.politi/Reprints/145.pdf
 | url-status = dead
 | archive-date = 2008-10-31
 | journal = Physical Review Letters
 | doi =  10.1103/PhysRevLett.99.130601
 | pmid = 17930570
 | issue = 13
| bibcode=2007PhRvL..99m0601G|arxiv = 0706.0510 |hdl=2158/253565 |s2cid=21992110 }}

===Software===
* [https://web.archive.org/web/20161027044059/http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/index.html] R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, [[Tisean|TISEAN]] 3.0.1 (March 2007).
* [https://web.archive.org/web/20130917012451/http://www.scientio.com/Products/ChaosKit] Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided online via a web service and Silverlight demo.
* [http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz] {{Webarchive|url=https://web.archive.org/web/20220628200022/http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz |date=2022-06-28 }} Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval. The [http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c/ReadMe.html contents and manual pages] of the mathrec software laboratory are also available.
* [http://biocircuits.ucsd.edu/pbryant/] Software on this page was developed specifically for the efficient and accurate calculation of the full spectrum of exponents.  This includes LyapOde for cases where the equations of motion are known and also Lyap for cases involving experimental time series data.  LyapOde, which includes source code written in "C", can also calculate the conditional Lyapunov exponents for coupled identical systems.  It is intended to allow the user to provide their own set of model equations or to use one of the ones included.  There are no inherent limitations on the number of variables, parameters etc.  Lyap which includes source code written in Fortran, can also calculate the Lyapunov direction vectors and can characterize the singularity of the attractor, which is the main reason for difficulties in calculating the more negative exponents from time series data.  In both cases there is extensive documentation and sample input files.  The software can be compiled for running on Windows, Mac, or Linux/Unix systems.  The software runs in a text window and has no graphics capabilities, but can generate output files that could easily be plotted with a program like excel.