Universal Properties and Universal Numbres and their Measurments in Experiments on Chaotic Dynamical Systems*.

Abstract

The number of carefull experiments that exhibit a transition to low dimensional chaotic motion has been growing steadilly in recent years.1 Techniques to characterize strange attractors and to measure important invariants like dimensions 2,3, entropies 4,5 and Lyapunov exponents were prposed and widely implemented.There has not been however sufficient experimental stress on universal properties. To my knowledge, the only experimental verifications of universal behviour to date pertained to the subcritical regime (on the way to chaos) and right at threshold. These include measurements of the accumulation rate of period doublings, the spectrum at onset of chaos via period doubling, and recently some universal numbers and spectra pertaining to the transition via quasiperiodicity.8 In the chaotic regime one is still coming up with numbers for the invariants that are not related to any universal properties.In this lecture I plan to review the recent theoretical results on universal behaviour in the chaotic regime, in addition to some novel methods for seeing universality at threshold. Most of these predictions have not been corroborated experimentally yet.