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Philosophy of Science

Do Attractors Exist in Physical Space: The Truth Is Out There

Assen I. Dimitrov
Bulgarian Academy of Sciences

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The crossword

Suppose we have an arbitrary sequence of words without any semantic or syntactical correlation. (In an even simpler case we may have just an arbitrary sequence of letters.) We may vainly try to find whatever invariant dependencies within the above word chain. For example we may try whether the words are not composed of vowels and syllables according to a fixed algorithm. At best we may find that some symbols recur. Perhaps more often than in a really arbitrary word string? Perhaps not? Or, perhaps, we may also try to interpret the sequence as an encoded message? Again, without success. Shall we give up? - Not necessarily. - It may simply turn out that the word chain is a set of numerically ordered 'values' appearing in a crossword.

In point of fact we can find even linear dependencies within the considered, and within any arbitrary, long enough, word string. There will always be recurring symbols, which allow to transform any long enough word series into a crossword. It is another question that it will be a 'bad' crossword. Bad crosswords have too many black grids. Yet any large crossword has them.

But does the crossword physically exist? In particular, does it exist as an invariant physical dependency in the reality beyond the newspaper pages? - No, the reply must be categorically negative, be it just for our vain efforts to find any linear algorithm about it.

The attractor

Now suppose, we have a long and obscure series of numerical data. Basing ourselves on time delay state space reconstruction methods, we may find out that there is (or there is not) an attractor which underlies the series. - If we add an additional dimension to our linear word chain, it turns into a regular, organized structure - the crossword. So does the attractor. If we constitute a relevant state space - the attractor as a stable structure will emerge. - Otherwise, if we assume a 0/3-dimensional real space Euclidean standard - it won't, or possibly could in rare cases. (Despite that the raw time series data remains virtually the same before and after any coordinate transformation.)

What is essential about phase space is not only the (possibly different from 3) number of coordinates, but their qualitative nature. They rarely coincide with what we call spatial length, width and height. Most often the coordinates (or degrees of freedom) are physical variables devoid of any dimension.

But does the attractor exist? - The one possibility is that it may be just a pure mathematical construction, albeit of incomparably higher degree of complexity (compared to the crossword). On the other hand, no one takes crosswords seriously, or applies them to, say, weather forecasting. While whenever meteorological data represented in a state space converge to the domain of a strange attractor, we do expect sensitive instabilities in the atmosphere.

Meteorological data cannot be analyzed on a linear scale. The stable model (solution) is obtained only in abstract multidimensional phase space. Whenever we have a nonlinear real time series, very often there also exist:

  • its stable model (in abstract state space);
  • but also, the objective physical dependencies between its variables.

It is the objective physical dependencies that render the weather (in)stable. Thus, the stability conditions exist in reality, yet they can be revealed only in abstract state space. Do then, the stability conditions influence weather only provided they are correlated objectively, precisely as they are abstractly correlated in theoretical state space?!

The question is not trivial. It is whether the weather sensitive correlation is simply revealed when its degrees of freedom are represented in a multidimensional state space. Or perhaps this driving correlation must also exist and be valid as an event in real space?

Remember our crossword. It cannot exist as a true crossword in any other space, but planar. It is hard to imagine that someone could solve it as a raw linear word string. And if he could, this will be only because he can think of the series as two-dimensional. The crossword correlations - regardless whether on a paper, or just as in our head can only be two-dimensional. Or - not be at all.

To come to the attractor - it can fold up only in a x-dimensional state space. Or - simply - not fold. Yes, the state space may be abstract, an artifact, too. But it is the only space in which the attractor can fold. And if it doesn't converge, just the raw series will exist. Which is of no more use than a dismantled car... An objectively driving correlation could exist only in space identical with the abstract multidimensional phase space. Or not exist at all. There is no such driving correlation within the raw series itself, and no reason for fluctuating behaviour in it either.

But still... We construct an ideal state space out of variables we consider essential for weather forecasting. We further extrapolate the behaviour of an ideal point for freely chosen values of the ruling parameters. The set of possible solutions for the point's evolution, i.e. the set of its trajectories in state space is really trapped into the 'black hole' of the attractor. But just our ideal probe, the ideal trajectory evolving in the abstract state space. Not the atmospheric mass. It was only a game... If it was really possible to create an object composed out of say seven vector parameters evolving along a pre-established rule, then it would have really collapsed into the attractor sink. But there is no such an object in reality. Our specifically chosen and ordered set of numbers does land onto an attractor. But not the original scalar measurement time series. Nor the atmospheric mass itself. They just remain raw - respectively - a time series and a dynamical system.

Is this the final judgement? - Not again. Because even in most authoritative publications on chaos we see the same contradiction recurring over and over again. It is spoken of 'nonlinear time series'. If the time series is nonlinear, then it is already regarded as displayed in multidimensional state space. So it is no more a manifestation of raw reality. Then we should perhaps change the label to 'scalar time series'. If so, the series could really be considered 'unmanipulated' (authentic). But then, it also will be incapable of fluctuating, because fluctuations can occur only within 'manipulated', phase space. As crosswords could appear only in newspapers. (1)

(1) The same problem occurs with respect to human perceptual and cognitive capacity. The multidimensional representation of the environmental time series impinging on our senses yields dependencies unobservable within the linear scale. The series thus becomes a visible (coloured), spatio-temporal and a conceivable object. Properly organized data inevitably land onto relevant attractors and give rise to the (intuitively familiar, yet ontologically mysterious) mental states.

But this happens at the conscious individual level. Here abstract multidimensional space = consciousness is an available prerequisite. - What about the evolutionary scale? Assume there was time without conscious observers, capable of designing multidimensional abstract phase spaces. How did they emerge? In this case multidimensional phase space is not a conscious atrifact, but a prerequisite for mentality. Should we regard this as an evidence for the objective origin, hence - character of (abstract?) multidimensional state space?

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Alligood Kathleen T., Sauer Tim D., Yorke James S. Chaos: An Introduction to Dynamical Systems. Springer, New York, etc., 1997

Kantz H., Schreiber T. Nonlinear Time Series Analysis. Cambridge University Press, 1997

Nicolis J.S. Dynamics of Hierarchical Systems. Spinger, New Yokr, etc., Heidelberg, 1986

Peitgen Heinz-Otto, Jurgens H, Saupe D. Chaos and Fractals: New Frontiers of Science. Springer, New York, etc. 1992

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