The Characterization of Fractal Measures as Interwoven Sets of Singularities: Global Universality at the Transition to Chaos

The most dramatic event in the development of the modern theory of the onset of chaos in dynamical systems has been the discovery of universality [1]. Especially well known are the universal numbers α and δ, which in the context of period doubling pertain to the universal scaling properties of the 2∞ cycle near its critical point, and the rate of accumulation of pitchfork bifurcations in parameter space respectively [1], This type of universality is however local, being limited to behavior in the vicinity of an isolated point either in phase space or in parameter space. In this paper I wish to review some recent progress in elucidating the globally universal properties of dynamical systems at the onset of chaos. This progress has been achieved in collaboration with M.H. JENSEN, A. LIBCHABER, L.P. KADANOFF, T.C. HALSEY, B. SHRAIMAN and J. STAVANS [2-4]. In “global universality” we mean that an orbit in phase space has metric universality as a whole set or that a whole range of parameter space can be shown to have universal properties [5]. Examples that have been worked out recently include the 2∞ cycle of period doubling, the orbit on a 2-torus with golden-mean winding number at the onset of chaos and the complementary set to the mode-locking tongues in the 2-frequency route to chaos. The approach used is however quite general, as will become apparent below.