Scaling of chaos versus periodicity: how certain is it that an attractor is chaotic?

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 ε can convert an attractor from chaotic to nonchaotic or vice versa. We call a parameter value where this can happen ε uncertain. The probability that a random choice of the parameter is ε uncertain commonly scales like a power law in ε. 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.