This article specifies a standard formal definition of chaos and comments briefly on the relationship of mathematical chaos to processes within the real physical world.
Frequently, chaos is defined just as sensitive dependence on initial conditions, but while sensitive dependence is a necessary property of chaotic systems, it is not by itself sufficient. Here I use a standard definition which is due to Devaney (1988) and which can be applied either to continuous systems described typically by differential equations or to iterated mappings. The definition requires topological transitivity and a dense covering of state space with periodic points. The second of these is the simplest to understand: the set of periodic points in the state space of a chaotic system at any given time slice, or Poincaré section, is at least countably infinite, and between any two points in the set there exists another also in the set. Periodic points are just those which lie on closed phase trajectories.
Topological transitivity indicates that any given neighbourhood in the phase space of a chaotic system is eventually visited by the trajectory of some point lying in any other given neighbourhood. Consider a system mapping a metric space to itself and choose any two open bounded sets J and K in that space. Then the phase trajectory under that mapping of at least one point x in J intersects K in finite time, regardless of how small or how spatially separated J and K may be. Alternatively, given any two such sets, some finite phase trajectory always connects some point in J to some point in K.
These two properties together logically entail the ever-popular sensitive dependence on initial conditions (Banks et al. 1992). Sensitive dependence describes the case in which, for any given point in a system’s state space, some other point within an arbitrarily small distance of the first lies on a phase trajectory which diverges from that of the first. In other words, a system is sensitively dependent on initial conditions when there exists some distance D greater than zero such that for any point x in the system’s metric space and for any closed neighbourhood N of that space, there exists at least one point y in N such that within a finite time, the distance between x and y under the action of the system exceeds D. Note that this property occurs for every point within the system’s phase space, and a straightforward proof shows that every neighbourhood N of an x includes infinitely many such diverging points.
Chaos in the Real World
The above definitions apply to systems defined by systems of equations, and a word of caution is in order. Properly speaking, chaos is a property of dynamical models, of particular ways of describing things — and not necessarily of any underlying physical reality which those models may be used to describe. Moreover, while it may be rigorously proven that a given equation or set of equations defines a chaotic system, empirical evidence cannot extend that certainty from the chaotic model to the physical entity being modelled. Starting with empirical measurements, we might search for tell-tale signs of characteristics like sensitive dependence or quasi-periodicity or period doubling, but finite measurements can never demonstrate characteristics which mathematically demand infinite precision. For instance, empirical evidence obviously can never show that the temporal evolution of a real physical system is periodic for infinitely many distinct initial states or that for any initial state there will always be another arbitrarily similar to it which diverges from it over time. Similarly, contrary to some popular assertions made by people who should know better, finite empirical measurements can never reject the hypothesis that a particular trajectory is simply a closed orbit, rather than lying on a strange attractor. In general, it is no more possible to prove from observation that a real physical system is chaotic than it is possible to prove it is governed by exactly such and such a set of equations. (Indeed, as defined here, it is doubtful whether the word ‘chaotic’ applies to the real physical world at all. In a forthcoming book, Peter Smith explores the incongruity between the infinite intricacy which chaotic models necessarily exhibit and the fuzzy physical world, in which infinite intricacy is apparently impossible.) This should go without saying, but all too often one encounters misinformed assertions that the weather is chaotic or that a particular area of the brain is chaotic.
It isn’t unusual to hear that because of sensitive dependence, chaotic systems are somehow utterly unpredictable, that after some particular time t, all predictions of a system’s behaviour become useless. Yet the order of quantifiers in the above definition of sensitive dependence reveals the mistake: it is the case that for any point in a given neighbourhood of metric space, there is some time t by which the trajectory of some other point in the neighbourhood will have diverged from that of the first by at least D. But it just is not the case that there is some magical time t which guarantees divergence of all the points within the neighbourhood! As it happens, the magnitude of divergence in each dimension for nearby typical phase trajectories is proportional to e raised to the power of lt, where l (usually the Greek letter lambda) is the system’s Lyapunov exponent (and e, roughly 2.71828, is the natural constant). This simple proportion yields, for any specific finite horizon of prediction t and any desired error bound d, the minimal precision needed in an initial measurement to guarantee that after t our error will be less than d. This feature is often called the Shadowing Theorem.
The theorem guarantees that for a calculable cost in initial precision, we can ‘shadow’ any trajectory as closely as we wish and for as long a finite time as we wish. So, the presence of chaos does not, broadly speaking, render a system impossible to model or to predict.
Chaos and Computability
Nor does chaotic behaviour in general somehow render a system noncomputable or endow it with special non-algorithmic powers. Indeed, generally speaking, chaos and computability have utterly nothing to do with one another. (A separate set of short introductions covers computability as well as an exception to the irrelevance of chaos for questions of computability.) ‘Chaotic’ and ‘computable’ are not opposites, and statements like “it might not be computable, because it might be chaotic” do not make any sense.
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This article was originally published by Dr Greg Mulhauser on .on and was last reviewed or updated by