{"quality_controlled":"1","title":"Rigorous numerics for critical orbits in the quadratic family","main_file_link":[{"url":"https://arxiv.org/abs/2004.13444","open_access":"1"}],"oa_version":"Preprint","issue":"7","language":[{"iso":"eng"}],"intvolume":"        30","day":"31","article_type":"original","publisher":"AIP","year":"2020","date_published":"2020-07-31T00:00:00Z","_id":"8694","doi":"10.1063/5.0012822","author":[{"last_name":"Golmakani","full_name":"Golmakani, Ali","first_name":"Ali"},{"orcid":"0000-0003-2640-4049","full_name":"Koudjinan, Edmond","first_name":"Edmond","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","last_name":"Koudjinan"},{"last_name":"Luzzatto","first_name":"Stefano","full_name":"Luzzatto, Stefano"},{"last_name":"Pilarczyk","first_name":"Pawel","full_name":"Pilarczyk, Pawel"}],"abstract":[{"lang":"eng","text":"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)=a−x2. 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 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."}],"article_number":"073143","publication_status":"published","publication":"Chaos","extern":"1","type":"journal_article","date_updated":"2021-01-12T08:20:34Z","volume":30,"external_id":{"arxiv":["2004.13444"]},"status":"public","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","month":"07","date_created":"2020-10-21T15:43:05Z","citation":{"ieee":"A. Golmakani, E. Koudjinan, S. Luzzatto, and P. Pilarczyk, “Rigorous numerics for critical orbits in the quadratic family,” <i>Chaos</i>, vol. 30, no. 7. AIP, 2020.","mla":"Golmakani, Ali, et al. “Rigorous Numerics for Critical Orbits in the Quadratic Family.” <i>Chaos</i>, vol. 30, no. 7, 073143, AIP, 2020, doi:<a href=\"https://doi.org/10.1063/5.0012822\">10.1063/5.0012822</a>.","ama":"Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. Rigorous numerics for critical orbits in the quadratic family. <i>Chaos</i>. 2020;30(7). doi:<a href=\"https://doi.org/10.1063/5.0012822\">10.1063/5.0012822</a>","short":"A. Golmakani, E. Koudjinan, S. Luzzatto, P. Pilarczyk, Chaos 30 (2020).","apa":"Golmakani, A., Koudjinan, E., Luzzatto, S., &#38; Pilarczyk, P. (2020). Rigorous numerics for critical orbits in the quadratic family. <i>Chaos</i>. AIP. <a href=\"https://doi.org/10.1063/5.0012822\">https://doi.org/10.1063/5.0012822</a>","chicago":"Golmakani, Ali, Edmond Koudjinan, Stefano Luzzatto, and Pawel Pilarczyk. “Rigorous Numerics for Critical Orbits in the Quadratic Family.” <i>Chaos</i>. AIP, 2020. <a href=\"https://doi.org/10.1063/5.0012822\">https://doi.org/10.1063/5.0012822</a>.","ista":"Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. 2020. Rigorous numerics for critical orbits in the quadratic family. Chaos. 30(7), 073143."}}