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**Quote:**Concerning the idea of Lyapunov : what algorithm is needed to calculate this ? Can you give me some hint ? I wasn't familiar with this .

The algorithm for Lyapounov Team's calculation I obtained from the theory of doctorate of the own solemnity of SOLSYIN (a long abstract in English tb is available. He is practically a complete article! Details can be obtained of the complete version (immense) of the tesis although she is in Rumanian. But Reading the theory first in English is not so difficult to understand the text in Rumanian. )

A importance of the Lyapounov Time is that ...

**"The phenomenon of chaos appears even in one of the simplest problems of celestial **

mechanics: the restricted three-body problem. Close encounters with the perturbatrice

planet always induce such a chaotic behaviour for an asteroidal orbit. But

the phenomenon of chaos is more subtle, since it appears in motions totaly free of close

encounters. For the planar, circular, restricted three-body problem the Poincare

surface of section is a tool to distinguish between chaotic and regular motions. For the

general case, we have the Lyapounov exponents method.

...

In summary, the exponential divergence in time of the specific three types of errors (error in initial

conditions, approximation error and round-off error) limits the deterministic nature of

the final numerical solution." reference:

http://math.ubbcluj.ro/~sberinde/thesis/abstract.pdf Simulating some dozens or hundreds of virtual images of an asteroid and calculating the medium difference among their longitudes is possible to obtain Lyapounov Time ( the length of time for a dynamical system to become chaotic. The Lyapunov time reflects the limits of the predictability of the system. By convention, it is measured as the time for nearby trajectories of the system to diverge by e. )

Starting from the generated graph it is possible if to obtain the approximate data of the variables of the algorithm

The algorithm is: L=1/ ((ln(Di)-ln(Df))/(Tf-Ti))

In the graph below Di = initial distance, in this case is 1e-04. The final distance (Df) is 100. The difference of time so that the chaotic state is reached is the time to the graph reach the point that the distance is 100 (Tf), in the case 2300, less the initial time of the simulation (Ti) that is 2000 in the graph.

This gives a time of Lyapounov 21,7 years old, in other words, this is the time that not modeled forces and mistakes of data act on the previsibility of the orbit. In the case of the graph we can foresee with safety the evolution of the orbit of the asteroid to approximately 21,7 years.

For Excel to use the following format:

1 / ((LN (B1) - LN (A1)) / (C1-D1) )

(A1, B1, C1 and D1 is the cells with the data.)

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