On the Computational Capabilities of Several Models

We review some results about the computational power of several computational models. Considered models have in common to be related to continuous dynamical systems. 1 Dynamical Systems and Polynomial Cauchy Problems A polynomial Cauchy problem is a Cauchy problem of type { x′ = p(x, t) x(0) = x0 where p(x, t) is a vector of polynomials, and x0 is some initial condition. The class of functions that are solution of a polynomial Cauchy problem turns out to be a very robust class [14]. It contains almost all natural mathematical functions. It is closed under addition, subtraction, multiplication, division, composition, differentiation, and compositional inverse [14]. Actually, every continuous time dynamical system x′ = f(x, t) where each component of f is defined as a composition of functions in the class and polynomials can be shown equivalent to a (possibly higher dimensional) polynomial Cauchy problem [14]. This implies that almost all continuous time dynamical systems considered in books like [16], or [21] can be turned in the form of (possibly higher dimensional) polynomial Cauchy problems. For example, consider the dynamic of a pendulum x′′ + p sin(x) = 0. Because of the sin function, this is not directly a polynomial ordinary differential equation. However, define y = x′, z = sin(x), u = cos(x). A simple computation of derivatives show that we must have  x′ = y y′ = −pz z′ = yu u′ = −yz , which is a polynomial ordinary differential equation. ? This work has been partially supported by French Ministry of Research through ANR Project SOGEA. This class of dynamical systems becomes even more interesting if one realizes that it captures all what can be computed by some models of continuous time machines, such as the General Purpose Analog Computer (GPAC) of Shannon [25].