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