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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 ...
On 5 January 1975, the 12-bit field that had been used for dates in the TOPS-10 operating system for DEC PDP-10 computers overflowed, in a bug known as "DATE75". The field value was calculated by taking the number of years since 1964, multiplying by 12, adding the number of months since January, multiplying by 31, and adding the number of days since the start of the month; putting 2 12 − 1 ...
Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general, indicate stability, and that a positive largest Lyapunov exponent does not, in general, indicate chaos. Therefore, time-varying linearization requires additional justification. [4]
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
Lyapunov theory, a theorem related to the stability of solutions of differential equations near a point of equilibrium; Lyapunov central limit theorem, variant of the central limit theorem; Lyapunov vector-measure theorem, theorem in measure theory that the range of any real-valued, non-atomic vector measure is compact and convex
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The planets' orbits are chaotic over longer time scales, in such a way that the whole Solar System possesses a Lyapunov time in the range of 2~230 million years. [3] In all cases, this means that the positions of individual planets along their orbits ultimately become impossible to predict with any certainty.