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
in a ``chaotic set'', enclosure methods are the only
practically available approach, unless a first integral of the ODEs is
known; and
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 phenomena always requires 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.He 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 at u=-1/2h. Thus
the map is not a diffeomorphism, whereas the h-flow of any continuous
dynamical system is a diffeomorphism.