25 research outputs found
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
Frequency locking in the injection-locked frequency divider equation
We consider a model for the injection-locked frequency divider, and study
analytically the locking onto rational multiples of the driving frequency. We
provide explicit formulae for the width of the plateaux appearing in the
devil's staircase structure of the lockings, and in particular show that the
largest plateaux correspond to even integer values for the ratio of the
frequency of the driving signal to the frequency of the output signal. Our
results prove the experimental and numerical results available in the
literature.Comment: 20 figures, 2 figure
Bifurcation curves of subharmonic solutions
We revisit a problem considered by Chow and Hale on the existence of
subharmonic solutions for perturbed systems. In the analytic setting, under
more general (weaker) conditions, we prove their results on the existence of
bifurcation curves from the nonexistence to the existence of subharmonic
solutions. In particular our results apply also when one has degeneracy to
first order -- i.e. when the subharmonic Melnikov function vanishes
identically. Moreover we can deal as well with the case in which degeneracy
persists to arbitrarily high orders, in the sense that suitable generalisations
to higher orders of the subharmonic Melnikov function are also identically
zero. In general the bifurcation curves are not analytic, and even when they
are smooth they can form cusps at the origin: we say in this case that the
curves are degenerate as the corresponding tangent lines coincide. The
technique we use is completely different from that of Chow and Hale, and it is
essentially based on rigorous perturbation theory.Comment: 29 pages, 2 figure
Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality
On the nature of space fluctuations of solutions of dissipative partial differential equations
In this work we have analysed the nature of space fluctuations in dissipative Partial Differential Equations (PDEs). By taking a well known and much investigated dissipative PDE as our representative, namely the Swift–Hohenberg Equation, we estimated in an explicit manner the values of the crest factor of its solutions. We believe that the crest factor, namely the ratio between the sup-norm and the L2 norm of solutions, is a suitable and proper measure of space fluctuations in solutions of dissipative PDEs. In particular it gives some information on the nature of “soft” and “hard” fluctuations regimes in the flows of dissipative PDEs