Szebehely [79] quotes Poincaré's definition of a non-integrable system as one in which ``no analytical and globally valid integrals (invariant relations between variables) exist besides energy, momentum, and center of mass.'' He notes several formal ways of recognizing a chaotic system. Using frequency analysis, for example, chaotic motion is associated with a broad spectrum while periodic motion is associated with peaks. Poincaré's surface-of-section method turns periodic orbits into points, quasi-periodic orbits into lines, and chaotic motion into an irregular distribution of points.

Skeel [70] notes that a probabilistic formulation of chaos in the scope of molecular dynamics spreads an initially small ensemble into a uniform distribution via mixing. He cautions that the backwards error idea of interpreting numerical trajectories as being exact solutions to a nearby problem is not foolproof: first, for backward error to be a reasonable interpretation, it is necessary to ensure that numerical errors are much smaller than modelling errors. Second, backward error analysis should never be offered as more than a partial justification: it is still necessary to show, for example, that errors in the force field do not seriously effect the quantities of interest.

A very interesting, but overly pessimistic, paper is one by
Adams *et al. * [2]. They note that a numerical
solution to an ODE *always* remains close to a true solution
of the problem for *some* length of time. The interesting
quantity is the typical length scale of each of these sections of
the numerical solution. This is similar to the idea of
*shadow segments* discussed by Hayes [33]. The
points at which the numerical solution switches which true
solution it is following they define as a point of *diversion*.
This is similar to the idea of *glitches* in the shadowing
literature, and an area where a diversion/glitch is likely to
occur they call a *perturbation sensitive* neighborhood.
The phenomenon of numerical solutions encountering
glitches/diversions is termed *computational chaos*.
They argue that diversions may be detectable if some first
integral is violated at that point; however, as was shown in
the errata to Hayes [33], glitches in *N*-body
solutions, for example, do not appear to be detectable via
violations of energy conservation alone.

One very interesting observation made by Adams *et al. * concerns
integration of the Lorenz equations using a ``classical Runge-Kutta
method'' (they do not specify exactly which one). Using enclosure
techniques, they prove the existence of a periodic orbit, and then
study the occurrence of diversions when using the RK method to integrate
the orbit. They find that the orbit is periodic to graphical accuracy
for all time when using a timestep of *h*=1/256, and also for
*h*=1/64. However, for *h*=1/128 they find that the orbit diverges
drastically from the periodic orbit. ``This unexpected non-monotonic
dependence of the accuracy of the solution on *h*,'' they emphasize,
``is *unpredictable*''. They discuss several other examples of
unpredictable behaviour of a numerical solution in comparison with the
solution produced by enclosure methods. Based on these results, they
question whether true solutions of the Lorenz equations, or even of
celestial mechanical systems that are generally believed to be chaotic
[81, 78, 52],
are in fact chaotic. They further use this
result to claim that shadowing results for ODEs are practically useless
since, they (erroneously) claim, there is no iron-clad guarantee of a
bound on local error. This is simply not true for systems for which
the right hand side of the ODE is sufficiently smooth
[30]. For example, Grebogi *et al. * [28] build a
shadow for a numerical solution of the forced, damped pendulum equation,
which has rigorous, explicit bounds on truncation error. Finally,
Adams *et al. * come to the astonishingly pessimistic (and wrong) conclusions
that

**(I)**- Concerning quantitatively reliable information on individual true solutions in a ``chaotic set'', enclosure methods are the only practically available approach, unless a first integral of the ODEs is known; and
**(II)**- An unknown but presumably large portion of the published
results on ``chaotic sets of solutions to ODEs'' is more concerned with
computational chaos than with ODE-chaos.

Corless [19] argues that perhaps too much emphasis is put on exponential magnification of numerical errors. He bases this conclusion on the observation that even real systems have perturbations that, for practical and even fundamental purposes, we necessarily ignore[19, p.32,]:

Mathematical modelling of real phenomenaHe does note, however, that there are sometimes fundamental problems when trying to imbed a discrete solution into a continuous system. For example, Euler's method applied to produces , which has derivative 0 atalwaysrequires approximation and neglect of small effects. One neglects, for example, the effect of the gravitational attraction of Jupiter on one's earthbound experiment.... Similarly, one ignores `small' stochastic terms in ordinary differential equation models of many phenomena, or `small' non-autonomous perturbations of physics experiments (such as the effect of passing trucks). So a numerical analysis of methods of solving ODEs which puts truncation and roundoff errors on the same basis as modelling, measurement, and data errors would be a completely successful analysis.... We [have to] study the effects of perturbations, of course, but we have to do this even if we know the exact solution of the specified problem.

Fri Dec 27 17:41:39 EST 1996

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