190 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
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 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 , 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
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