Theory of the Intermittent Chaos: 1/f Spectrum and the Pareto-Zipf Law

Abstract

The statistical properties of the intermittent chaos are investigated for a simple one dimensional map approximating the Lorenz model. We consider two kinds of intermittent behaviors which respectively occur after the destabilization of a periodic motion and a non-mixing island chaos. The residence time distribution of the laminar state is calculated numerically and we get the algebraic distribution. The 1/f power spectrum with fine oscillations is theoretically studied for the {Pomeau-Manneville-type} intermittent chaos. The effect of the small external additive noise is also investigated and the stability of the {Pareto-Zipf} law and the 1/f spectrum are explained by the renewal process analysis.