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Lyapunov time

Timescale of dynamical systems

Timescale of dynamical systems

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.

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. 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.

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.

Examples

Typical values are:

SystemLyapunov time
Pluto's orbit20 million years
Solar System5 million years
Axial tilt of Mars1–5 million years
Orbit of 36 Atalante4,000 years
Rotation of Hyperion36 days
Chemical chaotic oscillations5.4 minutes
Hydrodynamic chaotic oscillations2 seconds
1 cm3 of argon at room temperature3.7×10−11 seconds
1 cm3 of argon at triple point (84 K, 69 kPa)3.7×10−16 seconds

References

References

  1. (5 September 2010). "Extracting Knowledge from Time Series: An Introduction to Nonlinear Empirical Modeling". Springer.
  2. (2018). "Isomorphism between Maximum Lyapunov Exponent and Shannon's Channel Capacity".
  3. (2001). "A Comparison Between Methods to Compute Lyapunov Exponents". The Astronomical Journal.
  4. (2009). "On the Numerical Computability of Asteroidal Lyapunov Times".
  5. Gaspard, Pierre. (2005). "Chaos, Scattering and Statistical Mechanics". [[Cambridge University Press]].
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