1. Rigorous numerics for critical orbits in the quadratic family.
- Author
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Golmakani, A., Koudjinan, C. E., Luzzatto, S., and Pilarczyk, P.
- Subjects
CRITICAL point (Thermodynamics) ,PHASE space ,WEBSITES ,DYNAMICAL systems ,MAGNITUDE (Mathematics) - 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 P of the parameter interval Ω = [ 1.4 , 2 ] into almost 4 × 10 6 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 ω ∈ P + 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. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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