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

TL;DR

This paper investigates how small parameter changes can switch a system's attractor between chaotic and non-chaotic states, revealing different scaling behaviors depending on the definition used, and provides a detailed analysis of this phenomenon.

Contribution

It explains why two similar scaling definitions produce different exponents and offers a quantitative analysis of the scaling behavior of chaos versus periodicity.

Findings

Different definitions of $oldsymbol{ ext{scaling exponent}}$ yield different numerical values.

The probability of $oldsymbol{ ext{parameter uncertainty}}$ scales as a power law with $oldsymbol{ ext{perturbation size}}$.

The paper provides a detailed explanation of the causes behind the differing scaling exponents.

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