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. It is defined as the inverse of a system's largest Lyapunov exponent.[1]
Use
editThe 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. 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][3]
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.[4][5]
Examples
editTypical values are:[2]
| 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 | 36 days |
| Chemical chaotic oscillations | 5.4 minutes |
| Hydrodynamic chaotic oscillations | 2 seconds |
| 1 cm3 of argon at room temperature | 3.7×10−11 seconds |
| 1 cm3 of argon at triple point (84 K, 69 kPa) | 3.7×10−16 seconds |
See also
editReferences
edit- ^ Bezruchko, Boris P.; Smirnov, Dmitry A. (5 September 2010). Extracting Knowledge from Time Series: An Introduction to Nonlinear Empirical Modeling. Springer. pp. 56–57. ISBN 9783642126000.
- ^ a b Pierre Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, 2005. p. 7
- ^ Friedland, G.; Metere, A. (2018). "Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity". arXiv:1706.08638 [cond-mat.stat-mech].
- ^ Tancredi, G.; Sánchez, A.; Roig, F. (2001). "A Comparison Between Methods to Compute Lyapunov Exponents". The Astronomical Journal. 121 (2): 1171–1179. Bibcode:2001AJ....121.1171T. doi:10.1086/318732.
- ^ Gerlach, E. (2009). "On the Numerical Computability of Asteroidal Lyapunov Times". arXiv:0901.4871 [physics.comp-ph].