...complete

I.e.,  the author is under no illusions about the remaining extent of his ignorance.

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...extracted.

That is, no finite list of statistics completely describes a full phase-space trajectory.

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...map

Remark: In one dimension, this is synonymous with uniformly hyperbolic, except there are no expanding directions.

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...i+1.

  In other words, let

 

be the first-order ODE. Note that is the solution of (9) using as the initial condition and integrating to time i+1. Then is the Jacobian of . The Jacobian measures how changes if is changed by a small amount. The resolvent is the integral of J(t) along the path , and describes how a small perturbation from at time gets mapped to a perturbation from at time . That is, is the solution of the variational equation

where I is the identity matrix. The reason the arguments to R seem reversed is for notational convenience: they satisfy the identity , and so a perturbation at time gets mapped to a perturbation at time by the matrix-matrix and matrix-vector multiplication [30]. Finally, the linear map in the GHYS refinement procedure is .

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...them.

For any system, even a chaotic one, given any true orbit of fixed length, a small enough perturbation in the initial condition in any direction produces a small change in the final condition, although for chaotic systems this perturbation must be exponentially small in the length of the orbit. (If the perturbation is restricted to the stable subspace, then obviously a similar solution will be obtained.) Thus given any true orbit that -shadows a noisy orbit, there exist infinitely many true orbits nearby that also shadow it. However, it may be that all the true orbits are packed into a space unresolved by the machine precision.

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...possible.

  It is not hard to show that the Jacobian of the N-body problem has the form

where is symmetrical, and has real but not necessarily positive eigenvalues. If is an eigenvalue of , then are eigenvalues of J. If is positive, this gives one expanding and one contracting direction; if is negative, it gives directions that neither expand nor contract, but ``rotate'' in some sense. These are the directions that are non-hyperbolic.

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....

We may need to normalize the latter as in case or gets too small.

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...endpoints:

We will ignore the rescaling of time for this section, because I don't understand enough about it yet.

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...not.

Galactic clusters of stars and areas in the spiral arms perturbed by giant molecular clouds can have substantially higher local collision rates. Both these effects are negligible in elliptical galaxies, because ellipticals are hotter (and are thus less likely to contain clusters) and are generally cloudless.

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