# Lyapunov time

In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. See the extensive discussion of the Lyapunov exponent, its inverse.[1]

## Use

The Lyapunov time reflects 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. 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.[2]

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 stability of the Solar System question. However, empirical estimation of the Lyapunov time is often associated with computational or inherent uncertainties.[3][4]

## Examples

Typical values are:[2]

System Lyapunov time
Solar system 50 million years
Pluto's orbit 20 million years
Obliquity of Mars 1-5 million years
orbit of 36 Atalante 4,000 years
Rotation of Hyperion 36 days
Chemical chaotic oscillations 5.4 minutes
Hydrodynamic chaotic oscillations 2 seconds
1 cubic cm of argon at room temperature 3.7×10−11 seconds
1 cubic cm of argon at triple point 3.7×10−16 seconds

## References

1. ^ Boris P. Bezruchko, Dmitry A. Smirnov, Extracting Knowledge From Time Series: An Introduction to Nonlinear Empirical Modeling, Springer, 2010, pp. 56--57
2. ^ a b Pierre Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, 2005. p. 7
3. ^ G. Tancredi, A. Sánchez, F. ROIG. A comparison between methods to compute Lyapunov Exponents. The Astronomical Journal, 121:1171-1179, 2001 February
4. ^ E. Gerlach, On the Numerical Computability of Asteroidal Lyapunov Times, http://arxiv.org/abs/0901.4871