Rigorous numerics for critical orbits in the quadratic family

Abstract

We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We illustrate the effectiveness of our approach by constructing a dynamically defined partition $\mathcal P$ of the parameter interval $\Omega=[1.4, 2]$ into almost 4 million subintervals, for each of which we compute to high precision the orbits of the critical points up to some time $N$ and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide $\mathcal P$ into a family $\mathcal P^{+}$ of intervals which we call stochastic intervals and a family $\mathcal P^{-}$ of intervals which we call regular intervals. We numerically prove that each interval $\omega \in \mathcal P^{+}$ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in $\omega$ has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in $\mathcal P^{+}$ are stochastic and most parameters belonging to the intervals in $\mathcal P^{-}$ are regular, thus the names. We prove that the intervals in $\mathcal P^{+}$ occupy almost 90% of the total measure of $\Omega$. The software and the data is freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parametrized families of dynamical systems.