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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 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 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 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 is freely available at this http URL, 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.
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Authors (4)
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A. Golmakani
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C. E. Koudjinan
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S. Luzzatto
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- Copyright (2020 Author(s))
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- Category:
- Articles
- Type:
- artykuły w czasopismach
- Published in:
-
CHAOS
no. 30,
ISSN: 1054-1500 - Language:
- English
- Publication year:
- 2020
- Bibliographic description:
- Golmakani A., Koudjinan C., Luzzatto S., Pilarczyk P.: Rigorous numerics for critical orbits in the quadratic family// CHAOS -Vol. 30,iss. 7 (2020), s.073143-
- DOI:
- Digital Object Identifier (open in new tab) 10.1063/5.0012822
- Verified by:
- Gdańsk University of Technology
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