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 fa(x)=ax2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition P of the parameter interval Ω=[1.4,2] into almost 4×106 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 P into a family P+ of intervals, which we call stochastic intervals, and a family P of intervals, which we call regular intervals. We numerically prove that each interval ωP+ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in ω has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in P+ are stochastic and most parameters belonging to the intervals in P are regular, thus the names. We prove that the intervals in P+ occupy almost 90% of the total measure of Ω. The software and the data are 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 parameterized families of dynamical systems.

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