研究了一维二次映射族$f_a (x) = a - x^2$。我们进行全面的数值分析的收集有限轨道的临界点，计算的参数值的区间，使用严格的数值方法。我们使用计算机显式地构造包含在$Ω=[1.4, 2]$中的数千个参数区间的集合，这些参数区间被证明具有特定的所谓逃逸时间，这大致意味着在该区间中的所有参数上所取的临界点的一些有效计算的时间在相空间中具有相当大的宽度。特别是，我们计算一个严格的下限，这个宽度，除了上限。我们调查的效果施加在数值计算上的某些限制后，所产生的收集的间隔。此外，我们说明和讨论的参数空间中的计算间隔的分布。我们的工作的目的是建立进一步的数值计算的一个下界的随机参数在$Ω$的措施的理由。该软件的源代码和论文中讨论的数据可以在http://www.pawelpilarczyk.com/quadr/上免费获得。该网页还允许执行一些有限的计算。文中介绍的思想和程序可以很容易地推广到其他参数化的动力系统族。
英文摘要：
We study the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We conduct comprehensive numerical analysis of collections of finite orbits of the critical point, computed for intervals of parameter values using rigorous numerical methods. We use the computer to explicitly construct a collection of several thousand parameter intervals, contained in $Ω=[1.4, 2]$, that are proved to have a specific so-called escape time, which roughly means that some effectively computed iterate of the critical point taken over all the parameters in that interval has considerable width in the phase space. In particular, we compute a rigorous lower bound on this width, in addition to the upper bound. We investigate the effect of certain constraints imposed on the numerical computations upon the resulting collection of intervals. Additionally, we illustrate and discuss the distribution of the computed intervals in the parameter space. The purpose of our work is to establish grounds for further numerical computation of a lower bound on the measure of stochastic parameters in $Ω$. The source code of the software and the data discussed in the paper are freely available at http://www.pawelpilarczyk.com/quadr/. This web page also allows carrying out some limited computations. The ideas and procedures introduced in the paper can be easily generalised to apply to other parametrised families of dynamical systems.
