Scaling of Chaos versus Periodicity: How Certain is it that an Attractor is Chaotic?

Abstract

The character of the time-asymptotic evolution of physical systems can have complex, singular behavior with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter by a small amount can convert an attractor from chaotic to non-chaotic or viceversa. We call a parameter value where this can happen -uncertain. The probability that a random choice of the parameter is -uncertain commonly scales like a power law in . Surprisingly, two seemingly similar ways of defining this scaling, both of physical interest, yield different numerical values for the scaling exponent. We show why this happens and present a quantitative analysis of this phenomenon.While low-dimensional chaotic attractors are common and fundamental in a vast range of physical phenomenon, it is predominantly the case that chaotic motions in such systems are 'structurally unstable'. This typically has the consequence that an arbitrarily small change of a system parameter can always be found that results in periodic behavior [1]. Physical models displaying chaotic attractors that are structurally unstable arise very often, e.g., in studies of plasma dynamics [2], Josephson junctions [3], chemical reactions [4] and many others. We also note that even the simple example of the one-dimensional quadratic map,displays this phenomenon, and, in this paper we will study this example as a convenient paradigm for such situations in general.One reason for concern with this type of behavior is that physical systems have uncertainties in the values of their parameters, and one might therefore ask how confident one can be about the prediction of chaotic behavior from a model calculation (even when the model and its parameter dependence are precisely known and there is no noise). Despite the fundamental importance of this question, very little study has been done to quantitatively address it [5][6][7]. It is the purpose of this paper to readdress this general issue, and, in particular, to resolve a long-standing puzzle. This puzzle has to do with the scaling characterization of the fractal-like chaotic/periodic interweaving structure of parameter dependence associated with structural instability. In particular, studies on the quadratic map Eq.(1) have addressed scaling in two slightly different ways and obtained significantly different estimates of the scaling exponent [5][6][7]. The reason for this surprising discrepancy has remained unresolved. In one of the two ways of addressing scaling, attention was restricted to what might seem to be the most obvious source of uncertainty, namely, when a large (to be defined subsequently) chaotic attractor suddenly turns into a periodic attractor, as a parameter is varied. However, we find such transitions far too rare [5,7] to account for the observed uncertainty, applicable when all chaotic attractors of any size are considered [6], and this observation is at the heart of the resolution of the above-mentioned puzzle.Consider a dynamical system depending on a parameter C, and an attrac...

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