Universal Fine Structure of the Chaotic Region in Period-Doubling Systems

Abstract

A new relation is reported which quantitatively describes the fine structure of the chaotic region of period-doubling systems. The relation determines the onset of fundamental periods and of ergodic behavior. It involves bifurcation rates ${\ensuremath{\gamma}}_{k}$, which converge to a new universal constant $\ensuremath{\gamma}=2.94805\dots{}$. This theory is in agreement with numerical determinations of ${\ensuremath{\gamma}}_{k}$.