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 $Ω=[1.4, 2]$ 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 $\mathcal P$ 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 $ω\in \mathcal P^{+}$ 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 $\mathcal P^{+}$ 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 $Ω$. 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.
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 sp…
In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapp…
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).
In this discussion draft, we explore different duopoly games of players with quadratic costs, where the market is supposed to have the isoelastic demand. Different from the usual approaches based on numerical computations, the methods used in the present work are built on symbolic computations, which can produce analytical and rigorous results. Our investigations show that the stability regions are enlarged for the games considered in this work compared to their counterparts with linear costs, which generalizes the classical results of "F. M. Fisher. The stability of the Cournot oligopoly solu…
Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation
VU University Amsterdam（阿姆斯特丹自由大学）；Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Université Laval, Département de Mathématiques et de Statistique（拉瓦尔大学数学与统计系）
