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|t doesnÕt overlap a basin boundary. Since the distance from any point not on a boundary to the nearest boundary is trivially positive, [P*] can provide any e we require to satisfy [P**]. (Because for closed systems there is no escaping from attractor basins, for the moment we might just ignore the ("t) of [P**]; I include it to avoid needlessly broadening the class of other systems to which [P**] might apply.) Momentarily we will examine a system which illustrates that this line of reasoning is flawed.|
It is clear, of course, that the converse relationship does not hold: a systemÕs meeting [P**] in particular neighbourhoods does not imply its meeting [P*] in those same neighbourhoods; for a simple example where one is met but not the other, consider any system which has critical points on its strange attractor. In such a system [P*] might fail in the vicinity of the critical points; yet trajectories in the same neighbourhood could never leave the attractor and end up in a different basin, and [P**] would be satisfied.
Now if a given initial condition, an x0 from [P**], lies far from the basin of a boundary of attraction, then the Ôfinal stateÕ of the system can be predicted with certainty as long as our measurement error e is smaller than the distance from x0 to the basin boundary. Even if x0 is close to an ordinary (non-fractal) basin boundary (or, alternatively, if e is larger), such that some of the points within a particular e of x0 are actually in a different basin of attraction, this proportion of initial conditions with uncertain outcomes generally scales linearly with e. In terms of extrapolating this property from mathematical models to systems in the real world, incidentally, it is worth noting that [P**] can no more be true of physical systems in general than can [P1'] and [P*]. (This is because we can choose an x0 close enough to a basin boundary that quantum uncertainty prevents us shrinking e small enough that it doesnÕt overlap the boundary.) Where a modelÕs qualitative predictability in the style of [P**] becomes most interesting is in the neighbourhood of a fractal basin boundary.
In this case, the proportion of uncertain initial conditions within e of x0 scales nonlinearly with e, in a way that is usually related to the Hausdorff-Besicovitch dimension of the basin boundary. In a correlate of SmithÕs observation that chaotic systems demand exponentially more accurate initial measurements as the desired time interval for prediction is increased, initial measurement accuracy requirements for prediction of final state in terms of attractor basins can also be highly nonlinear. Grebogi and colleagues (1987), for instance, describe a model of a kicked double rotor where decreasing the initial error e by a factor of 1010 yields a decrease in the proportion of uncertain trajectories within e of x0 of only a factor of 10. But even with such high costs in initial data, such a system remains qualitatively predictable in the sense of [P**].
Recently, however, John C. Sommerer and Edward Ott (1993) have described a chaotic model for which [P**] fails but for which, curiously, [P*] does not. In the Sommerer and Ott system, we are faced not just with a highly nonlinear relationship between e and the proportion of uncertain points. The Ôfinal stateÕ of their system in terms of attractor basins is uncertain for every e > 0; moreover, this property holds not just on a limited fractal basin boundary, but over the entire phase space volume of the systemÕs two basins of attraction.
These basins of attraction are called riddled: an attractor basin B is riddled if it satisfies the following [R]:
[R] ("x0 Î B)("e > 0)($y0)((|x0 - y0| < e) & (y0 Ï B))
In other words, a basin is riddled if, for any initial state x0 in the attractorÕs basin, the set of points within any arbitrarily small distance e > 0 of x0 but not in the same basin as x0 has positive volume in phase space. The Sommerer and Ott model is the first example of a physical system (as opposed to just a mathematical mapping, as in Alexander, et al 1992) which possesses riddled attractors. The model describes a particle moving in the x-y plane with an acceleration given by the sum of three forces: the gradient of a symmetric scalar potential, linear friction opposite to the particleÕs direction of movement, and a periodic external force in the x-direction. They reduced the dimensionality of the system through a function that returned mappings of points on a PoincarŽ section at times in phase with the periodic external force.
The four-dimensional phase space of this simplified system possesses an invariant plane (along y = vy = 0) with a strange attractor. Within this invariant subspace, one Lyapunov exponent of typical trajectories is negative while the other is positive, thus indicating chaotic dynamics on the attractor, and normal to the invariant subspace both Lyapunov exponents are negative for typical trajectories. This implies that a set of points not in the invariant subspace, with nonzero phase space volume, is attracted to the chaotic attractor, but, as the authors point out, this does not rule out a dense set of atypical orbits on the attractor having a positive normal Lyapunov exponent. This latter condition allows for initial conditions arbitrarily near the invariant subspace to be repelled from it. These points may eventually find themselves in one of two repelling regions of the scalar potential and go to positive or negative infinity in y and vy, the y velocity component. Thus there is a second ÔattractorÕ at ±¥ along these two dimensions. Sommerer and OttÕs work is the first known model of a physical system with the three conditions shown by Alexander and colleagues to imply riddled basins of attraction: an invariant manifold in phase space with a strange attractor, negative Lyapunov exponents normal to typical orbits in the attractor, and a positive volume of initial conditions in any region of phase space attracted to a different attractor.
Although this is the first model of a physical system with riddled attractor basins to be studied, and while such systems are of course only a subset of all chaotic physical systems, Sommerer and Ott indicate that the conditions for such systems are Ôby no means so restrictive that riddled basins can be considered unnaturalÕ. (Sommerer and Ott 1993, p. 140) Indeed, apart from the symmetry of the scalar potential, the equations of the system are unremarkable, and the systemÕs riddled basins are a highly robust feature which do not disappear under a wide range of changes in control parameters. It seems unlikely that such systems will be rare in nature.
For our present discussion, the first immediately salient point about such systems is that riddled attractor basins are strictly incompatible with [P**]. This is straightforward: when [R] is true, there will be a y0 in a different basin of attraction to x0 for every e > 0. Yet [P**] would require some e > 0 which guaranteed that every y0 within e of x0 was in the same basin of attraction. It appeared originally that satisfaction of [P*] implied satisfaction of [P**], and so we might infer that systems with riddled basins fail [P*] as well. Yet while such systems are not Ôqualitatively predictableÕ, they do remain Ôepistemically deterministicÕ as we have defined the terms. The line of reasoning which suggested that [P*] implied [P**] was flawed in that it assumed there was some distance from the x0 trajectory to the nearest basin boundary; but in the case of riddled basins there are ÔholesÕ leading to other attractors in all neighbourhoods, and [P*] cannot provide a d to satisfy [P**] for a given x0. There is no indication that Sommerer and OttÕs system could somehow circumvent the Shadowing Theorem; thus we can only assume that along with a broad class of coupled nonlinear differential equations it does satisfy [P*].
Of course riddled basin systems are also still deterministic in the original [P1] sense: if the initial condition is known precisely, then the systemÕs exact final state can, in principle, be calculated without error for any future time. But if there is any imprecision whatsoever in our knowledge of the initial condition, we retain the ability to predict the systemÕs future state within d for any chosen amount of time, but we lose the ability to know the systemÕs ultimate behaviour in terms of attractor basins.
Thus, paradoxically, we can shadow the centre point of a neighbourhood to within an arbitrarily small distance, but we cannot comment on the destiny of other trajectories within that neighbourhood except up to the point in time to which we have already calculated. Essentially, we can keep trajectories within a certain area, but we cannot say where they are going! The best we could do would be to choose each of the points in the neighbourhood in turn and calculate their trajectories, one by one. In principle we can calculate the destiny of any single point in phase space, but we cannot infer anything about the long term destiny of points lying arbitrarily near it. Within the distance d of where we might know a particular point x0 will map after a given amount of time, there is qualitative variance in trajectories in that a positive volume of points within e of x0 will diverge from it sufficiently to go to a different attractor than x0 itself.
Recall that qualitative variance of trajectories within a small neighbourhood of a given point on a computable trajectory is exactly what I suggested earlier, on a priori grounds, should be possible for a class of analogue chaotic systems. We have observed already that due to problems of quantum measurement, our criterium for qualitative predictability is not met by actual physical systems which are chaotic. But we now have to hand an example of a mathematical model which also fails to meet the criterium. Unpredictability in the physical world was one thing, but in the mathematical world it is another. We have encountered a mathematical model governed by computable functions which displays an aspect of noncomputable behaviour.
It is specifically the continuous nature of the equations of motion Sommerer and Ott have exploited which yields exactly the kind of situation I earlier argued was possible. If there was only the denumerably infinite class of computable phase trajectories passing through points within a given e of x0 at a particular time, we could computably set about the task of calculating each of their ultimate destination in terms of attractor basin. But doing so gives us absolutely no information about the uncountably infinite class of noncomputable phase trajectories passing through the same space in an analogue system. For all we could know, the ÔrealÕ destination map of an analogue system with riddled attractors might spell out Ô© God, All Rights Reserved!Õ in noncomputable points in one attractor basin against a background of noncomputable points in another basin!
Indeed, a careful examination reveals that we cannot even be sure of the qualitative accuracy of the destination maps Sommerer and Ott include in their article. Regardless of the host of problems associated with the roundoff error in calculating the trajectories of points Sommerer and Ott chose, while we can be certain that those points which definitely arrive in the repellent area of the scalar potential will be pushed to infinity, we cannot be sure that any other points even arbitrarily close to the invariant plane will ultimately be attracted to it.20 Sommerer and Ott note this as the possibility of Ôarbitrarily long transientsÕ but fail to note its implication for the qualitative accuracy of their destination maps. It is tempting to think that the Shadowing Theorem guarantees that calculating the computable trajectories would give all the information we needed and that no trajectories in the neighbourhood would behave significantly differently, but we have already noted that truth of [P*] with respect to a given neighbourhood does not imply that [P**] is also true in that neighbourhood.
Of course the complexity of the Sommerer and Ott system is not restricted to questions of analogue systems: the dynamics of the system as run on a digital computer are highly complex as well! The specifics of the systemÕs behaviour can be reproduced only by digital computers with identical architectures and numerical evaluation algorithms. The authors note that their calculated destination maps of points in the PoincarŽ sections they studied differed in fine detail (but not general character) when they used different computers with different round-off algorithms and precision. This just reinforces the point that the details of behaviour for a digitally simulated system would likely be vastly different than for any analogue system.
Although I did not have [P*] and [P**] formulation to hand or even any knowledge of Sommerer and OttÕs research when I originally outlined the earlier arguments about computability of chaotic analogue systems, the new system suggests a way of quantifying one route to noncomputable behaviour in terms of [P*] and [P**]. Simply, it would appear that any analogue chaotic system described by computable functions and which fails [P**] but still meets [P*] will display the kind of noncomputable behaviour I have discussed. This is not to say there might not be other ways to get to similarly interesting behaviour, and the earlier arguments hint at what some of these might be; but this certainly appears to be one route to noncomputable system behaviour in the face of computable governing functions.
We have explored SmithÕs analysis of predictability in chaotic systems and seen some of the ways in which they do remain different than nonchaotic deterministic systems in general. Although the systems with [R] style basins of attraction, for which [P**] fails while [P*] remains true for typical trajectories, are the most striking examples of properties unique to chaotic systems, we have also noted how [P*] may fail in the vicinity of critical points and what impact this has on the systemsÕ predictability. The unique properties of chaotic systems are such that they cannot be quickly dismissed as irrelevant to cognitive modelling just because chaotic systems are in general Ôpredictable in principleÕ like other deterministic systems. These unique properties may have functionally relevant r™les in real intelligent systems which will not be served by any but the most detailed simulations of extremely fine physical details of neural networks present at or below the level of individual neurons.
Next we consider the problem of complexity and relating chaos to simple noise.
We now turn to SmithÕs last comment on chaotic models which is significant for our own project. The issue this time is whether chaotic behaviour can in some sense be categorised as random. A central question is whether the apparently chaotic behaviour we observe in the real world is qualitatively the same as the chaotic behaviour we observe in mathematical models. In the mathematical models, detailed chaotic behaviour might be viewed as the result of the equationsÕ nonlinearity magnifying the fine details of the expansions of real numbers used to specify a systemÕs initial conditions. In the real world, Smith would suggest, the detailed chaotic behaviour may be viewed as the result of the continual influence of environmental noise. We noted above, in the discussion about describing the behaviour of physical systems with mathematical models of lower dimensions, that injecting noise is a common and often indispensable tool for bringing the modelÕs behaviour in line with what is observed in the physical world. But if apparent chaos in the real world really is just the result of noise, then almost all philosophical questions we might ask about the importance of chaos