Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.Comment: 24 pages, 16 figure

Statistical Mechanical Theory of a Closed Oscillating Universe

Based on Newton's laws reformulated in the Hamiltonian dynamics combined with statistical mechanics, we formulate a statistical mechanical theory supporting the hypothesis of a closed oscillating universe. We find that the behaviour of the universe as a whole can be represented by a free entropic oscillator whose lifespan is nonhomogeneous, thus implying that time is shorter or longer according to the state of the universe itself given through its entropy. We conclude that time reduces to the entropy production of the universe and that a nonzero entropy production means that local fluctuations could exist giving rise to the appearance of masses and to the curvature of the space

Globally and locally attractive solutions for quasi-periodically forced systems

We consider a class of differential equations, x + yx + g(x) = f(omega t), with omega is an element of R-d, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of trajectories with the same quasi-periodicity as the forcing. For g(x) = x(2p+1), p is an element of N, we show that, when the dissipation coefficient is large enough, there is only one such trajectory and that it describes a global attractor. In the case of more general nonlinearities, including g(x) = x(2) (describing the varactor equation), we find that there is at least one trajectory which describes a local attractor

On the stability of the upside-down pendulum with damping

A rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position of the vertically driven pendulum, with a small nonvanishing damping term, becomes asymptotically stable when the period of the forcing is below an appropriate threshold value. As a by-product we obtain an analytic expression of the solution for initial data close enough to the equilibrium position