42 citations · 109 across the 13 of their papers we have counts for
14 papers
Robust non-computability of dynamical systems and computability of robust dynamical systems
Daniel S. Graça, Ning Zhong
In this paper, we examine the relationship between the stability of the dynamical system and the computability of its basins of attraction. We present a computabl…
Robustness of non-computability
Daniel S. Graça, Ning Zhong
Turing computability is the standard computability paradigm which captures the computational power of digital computers. To understand whether one can create physically realistic d…
Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines
Daniel S. Graça, Ning Zhong
In this paper, we analyze the problem of finding the minimum dimension such that a closed-form analytic map/ordinary differential equation can simulate a Turing machine over $\…
Computing the exact number of periodic orbits for planar flows
Daniel S. Graça, Ning Zhong
In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. U…
The set of hyperbolic equilibria and of invertible zeros on the unit ball is computable
Daniel S. Graça, Ning Zhong
In this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow defined on the closed unit disk, outputs the exact number…
Computing geometric Lorenz attractors with arbitrary precision
Daniel Graca, Cristobal Rojas, Ning Zhong
The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of \emph{strange attractors}. However Lorenz' research was mainly based on (non-rigourous)…
On the Functions Generated by the General Purpose Analog Computer
Olivier Bournez, Daniel Graça, Amaury Pouly
We consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continu…
Computing with Polynomial Ordinary Differential Equations
Olivier Bournez, Daniel Graça, Amaury Pouly
In 1941, Claude Shannon introduced the General Purpose Analog Computer(GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog…
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
Olivier Bournez, Daniel S. Graça, Amaury Pouly
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class of languages co…
An analytic system with a computable hyperbolic sink whose basin of attraction is non-computable
Daniel S. Graça, Ning Zhong
In many applications one is interested in finding the stability regions (basins of attraction) of some stationary states (attractors). In this paper we show that one cannot compute…
Computational complexity of solving polynomial differential equations over unbounded domains
Amaury Pouly, Daniel S. Graça
In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) over \emph{unbounded time domains}, where is a vec…
Turing machines can be efficiently simulated by the General Purpose Analog Computer
Amaury Pouly, Olivier Bournez, Daniel S. Graça
The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This…
On the complexity of solving initial value problems
Olivier Bournez, Daniel S. Graça, Amaury Pouly
In this paper we prove that computing the solution of an initial-value problem with initial condition at time with precision whe…
Computability, Noncomputability, and Hyperbolic Systems
Daniel S. Graca, Ning Zhong, Jorge Buescu
In this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point. These manifolds are the essential feature which characterizes a hyp…