Paper detail

The Scaling of Chaos vs 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 non-chaotic 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.