Resonance model for chaos near period three

Abstract

A detailed study of the complex dynamics generated by a quadratic one-dimensional map near period three is presented. A resonance model for the tangent bifurcation process is constructed, which leads to an expression for the probability density as a sum of three Lorentzian contributions whose widths are expressed in terms of the fixed points of the third iterates of the map in the complex plane. It is also shown that the coarse-grained dynamics near period three can be analyzed in terms of an intermittency picture where deterministic motion, localized at the density maxima, is interspersed with chaotic bursts. An analysis of this motion explains the square-root dependence of the computed Liapunov number on the deviation of the map parameter from its bifurcation value. The origin of the fine structure, which decorates the spikes in the probability density, is described. The applicability of this model to the chaotic flows of ordinary differential equations is also discussed with respect to the dynamics on the Rössler strange attractor.