In
mathematics, the Lyapunov time is the characteristic timescale on which a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
is
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids ...
. It is named after the
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...
n
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Aleksandr Lyapunov. It is defined as the inverse of a system's largest
Lyapunov exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with in ...
.
Use
The Lyapunov time mirrors the limits of the
predictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively.
Predictability and causality
Causal determinism has a strong relationship with predictability. Pe ...
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
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
where it is important for the problem of the
stability of the Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ...
. However, empirical estimation of the Lyapunov time is often associated with computational or inherent uncertainties.
Examples
Typical values are:
[Pierre Gaspard, ''Chaos, Scattering and Statistical Mechanics'', Cambridge University Press, 2005. p. 7]
See also
*
Belousov–Zhabotinsky reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element i ...
*
Molecular chaos
*
Three-body problem
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
References
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Dynamical systems