Rigorous numerics for critical orbits in the quadratic family
| Autor: | Stefano Luzzatto, Ali Golmakani, Paweł Pilarczyk, Comlan Edmond Koudjinan |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Discrete mathematics
Dynamical systems theory business.industry Applied Mathematics General Physics and Astronomy Parameterized complexity Statistical and Nonlinear Physics Dynamical Systems (math.DS) 01 natural sciences Critical point (mathematics) 010305 fluids & plasmas Quadratic equation Software Phase space 0103 physical sciences FOS: Mathematics Partition (number theory) Mathematics - Dynamical Systems 010306 general physics business Mathematical Physics Mathematics |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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