Rigorous computation of escape times for parameter intervals in the quadratic map

Abstract

We study the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We conduct comprehensive numerical analysis of collections of finite orbits of the critical point, computed for intervals of parameter values using rigorous numerical methods. We use the computer to explicitly construct a collection of several thousand parameter intervals, contained in $\Omega=[1.4, 2]$, that are proved to have a specific so-called escape time, which roughly means that some effectively computed iterate of the critical point taken over all the parameters in that interval has considerable width in the phase space. In particular, we compute a rigorous lower bound on this width, in addition to the upper bound. We investigate the effect of certain constraints imposed on the numerical computations upon the resulting collection of intervals. Additionally, we illustrate and discuss the distribution of the computed intervals in the parameter space. The purpose of our work is to establish grounds for further numerical computation of a lower bound on the measure of stochastic parameters in $\Omega$. The source code of the software and the data discussed in the paper are freely available at http://www.pawelpilarczyk.com/quadr/. This web page also allows carrying out some limited computations. The ideas and procedures introduced in the paper can be easily generalised to apply to other parametrised families of dynamical systems.