Editing Lyapunov time

{{Short description|Timescale of dynamical systems}}
In [[mathematics]], the '''Lyapunov time''' is the characteristic timescale on which a [[dynamical system]] is [[chaos theory|chaotic]]. It is named after the [[Russia]]n [[mathematician]] [[Aleksandr Lyapunov]].  It is defined as the inverse of a system's largest [[Lyapunov exponent]].<ref>{{cite book |first1=Boris P. |last1=Bezruchko |first2=Dmitry A. |last2=Smirnov |url=https://books.google.com/books?id=li6JDAEACAAJ |isbn=9783642126000 |publisher=Springer |pages=56–57 |title=Extracting Knowledge from Time Series: An Introduction to Nonlinear Empirical Modeling |date=5 September 2010 }}</ref>

==Use==
The Lyapunov time mirrors the limits of the [[predictability]] of the system. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of ''[[e (mathematical constant)|e]]''. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or one digit of precision respectively.<ref name="gaspard" /><ref>{{cite arXiv |eprint=1706.08638 |last1=Friedland |first1=G. |last2=Metere |first2=A. |title=Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity |year=2018 |class=cond-mat.stat-mech }}</ref>

While it is used in many applications of dynamical systems theory, it has been particularly used in [[celestial mechanics]] where it is important for the problem of the [[stability of the Solar System]]. However, empirical estimation of the Lyapunov time is often associated with computational or inherent uncertainties.<ref>{{cite journal |doi=10.1086/318732|title=A Comparison Between Methods to Compute Lyapunov Exponents|year=2001|last1=Tancredi|first1=G.|last2=Sánchez|first2=A.|last3=Roig|first3=F.|journal=The Astronomical Journal|volume=121|issue=2|pages=1171–1179|bibcode=2001AJ....121.1171T|doi-access=free}}</ref><ref>{{cite arXiv |eprint=0901.4871 |last1=Gerlach |first1=E. |title=On the Numerical Computability of Asteroidal Lyapunov Times |year=2009 |class=physics.comp-ph }}</ref>

==Examples==
Typical values are:<ref name="gaspard">Pierre Gaspard, ''Chaos, Scattering and Statistical Mechanics'', Cambridge University Press, 2005. p. 7</ref>
{| class="wikitable"
|-
! System !! Lyapunov time
|-
| [[Pluto]]'s orbit|| 20 million years
|-
| [[Solar System]] || 5 million years
|-
| [[Axial tilt]] of [[Mars]] || 1–5 million years
|-
| Orbit of [[36 Atalante]] || 4,000 years
|-
| Rotation of [[Hyperion (moon)|Hyperion]] || 36 days
|-
| Chemical chaotic oscillations || 5.4 minutes
|-
| [[Hydrodynamic]] chaotic oscillations || 2 seconds
|-
| 1 cm<sup>3</sup> of [[argon]] at room temperature || 3.7×10<sup>−11</sup> seconds
|-
| 1 cm<sup>3</sup> of argon at triple point (84 K, 69 kPa) || 3.7×10<sup>−16</sup> seconds
|}

==See also==
* [[Belousov–Zhabotinsky reaction]]
* [[Molecular chaos]]
* [[Three-body problem]]

==References==
{{reflist}}

[[Category:Dynamical systems]]