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

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.Comment: 32 pages, 8 figures, 5 table

Intermittency and regularity issues in 3D Navier-Stokes turbulence

Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active `events' during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray's weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into `good' and `bad' intervals: on the `good' intervals solutions are bounded and regular, whereas singularities are still possible within the `bad' intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these `bad' intervals, lower bounds on the local energy dissipation rate and other quantities, such as \|\bu(\cdot, t)\|_{\infty} and \|\nabla\bu(\cdot, t)\|_{\infty}, are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for $n\geq 1$ are related to Scheffer's potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given `bad' interval, the energy has a lower bound that is decaying exponentially in time.Comment: 36 pages, 3 figures and 6 Table

Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension $d$, for the existence of a quasi-periodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasi-periodic solution is proved to describe a local attractor.Comment: 10 page

Basins of attraction in forced systems with time-varying dissipation

We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.Comment: 16 pages, 13 figures, 7 table