13 research outputs found
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
Conditions for positivity of solutions of a class of dissipative partial differential equations
Lindstedt series for perturbations of isochronous systems: a review of the general theory
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