Properties of invariant distributions and Lyapunov exponents for chaotic logistic maps

Abstract

Statistical scientists have recently focused sharp attention on properties of iterated chaotic maps, with a view to employing such processes to model naturally occurring phenomena. In the present paper we treat the logistic map, which has earlier been studied in the context of modelling biological systems. We derive theory describing properties of the 'invariant' or 'stationary' distribution under logistic maps and apply those results in conjunction with numerical work to develop further properties of invariant distributions and Lyapunov exponents. We describe the role that poles play in determining properties of densities' iterated distributions and show how poles arise from iterated mappings of the centre of the interval to which the map is applied. Particular attention is paid to the shape of the invariant distribution in the tails or in the neighbourhood of a pole of its density. A new technique is developed for this application. it enables us to combine 'parametric' information, available from the structure of the map, with 'nonparametric' information obtainable from numerical experiments.