1,534 research outputs found
Fractional Lindstedt series
The parametric equations of the surfaces on which highly resonant
quasi-periodic motions develop (lower-dimensional tori) cannot be analytically
continued, in general, in the perturbation parameter, i.e. they are not
analytic functions of the perturbation parameter. However rather generally
quasi-periodic motions whose frequencies satisfy only one rational relation
("resonances of order 1") admit formal perturbation expansions in terms of a
fractional power of the perturbation parameter, depending on the degeneration
of the resonance. We find conditions for this to happen, and in such a case we
prove that the formal expansion is convergent after suitable resummation.Comment: 40 pages, 6 figure
Converging Perturbative Solutions of the Schroedinger Equation for a Two-Level System with a Hamiltonian Depending Periodically on Time
We study the Schroedinger equation of a class of two-level systems under the
action of a periodic time-dependent external field in the situation where the
energy difference 2epsilon between the free energy levels is sufficiently small
with respect to the strength of the external interaction. Under suitable
conditions we show that this equation has a solution in terms of converging
power series expansions in epsilon. In contrast to other expansion methods,
like in the Dyson expansion, the method we present is not plagued by the
presence of ``secular terms''. Due to this feature we were able to prove
absolute and uniform convergence of the Fourier series involved in the
computation of the wave functions and to prove absolute convergence of the
epsilon-expansions leading to the ``secular frequency'' and to the coefficients
of the Fourier expansion of the wave function
Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles
Quasi-periodic cocycles with a diophantine frequency and with values in
SL(2,R) are shown to be almost reducible as long as they are close enough to a
constant, in the topology of k times differentiable functions, with k great
enough. Almost reducibility is obtained by analytic approximation after a loss
of differentiability which only depends on the frequency and on the constant
part. As in the analytic case, if their fibered rotation number is diophantine
or rational with respect to the frequency, such cocycles are in fact reducible.
This extends Eliasson's theorem on Schr\"odinger cocycles to the differentiable
case
Predictions of local ground geomagnetic field fluctuations during the 7-10 November 2004 events studied with solar wind driven models
The 7-10 November 2004 period contains two events for which the local ground magnetic field was severely disturbed and simultaneously, the solar wind displayed several shocks and negative <i>B<sub>z</sub></i> periods. Using empirical models the 10-min RMS and at Brorfelde (BFE, 11.67&deg; E, 55.63&deg; N), Denmark, are predicted. The models are recurrent neural networks with 10-min solar wind plasma and magnetic field data as inputs. The predictions show a good agreement during 7 November, up until around noon on 8 November, after which the predictions become significantly poorer. The correlations between observed and predicted log RMS is 0.77 during 7-8 November but drops to 0.38 during 9-10 November. For RMS the correlations for the two periods are 0.71 and 0.41, respectively. Studying the solar wind data for other L1-spacecraft (WIND and SOHO) it seems that the ACE data have a better agreement to the near-Earth solar wind during the first two days as compared to the last two days. Thus, the accuracy of the predictions depends on the location of the spacecraft and the solar wind flow direction. Another finding, for the events studied here, is that the and models showed a very different dependence on <i>B<sub>z</sub></i>. The model is almost independent of the solar wind magnetic field <i>B<sub>z</sub></i>, except at times when <i>B<sub>z</sub></i> is exceptionally large or when the overall activity is low. On the contrary, the model shows a strong dependence on <i>B<sub>z</sub></i> at all times
Energy localization on q-tori, long term stability and the interpretation of FPU recurrences
We focus on two approaches that have been proposed in recent years for the
explanation of the so-called FPU paradox, i.e. the persistence of energy
localization in the `low-q' Fourier modes of Fermi-Pasta-Ulam nonlinear
lattices, preventing equipartition among all modes at low energies. In the
first approach, a low-frequency fraction of the spectrum is initially excited
leading to the formation of `natural packets' exhibiting exponential stability,
while in the second, emphasis is placed on the existence of `q-breathers', i.e
periodic continuations of the linear modes of the lattice, which are
exponentially localized in Fourier space. Following ideas of the latter, we
introduce in this paper the concept of `q-tori' representing exponentially
localized solutions on low-dimensional tori and use their stability properties
to reconcile these two approaches and provide a more complete explanation of
the FPU paradox.Comment: 38 pages, 7 figure
A Cantor set of tori with monodromy near a focus-focus singularity
We write down an asymptotic expression for action coordinates in an
integrable Hamiltonian system with a focus-focus equilibrium. From the
singularity in the actions we deduce that the Arnol'd determinant grows
infinitely large near the pinched torus. Moreover, we prove that it is possible
to globally parametrise the Liouville tori by their frequencies. If one
perturbs this integrable system, then the KAM tori form a Whitney smooth
family: they can be smoothly interpolated by a torus bundle that is
diffeomorphic to the bundle of Liouville tori of the unperturbed integrable
system. As is well-known, this bundle of Liouville tori is not trivial. Our
result implies that the KAM tori have monodromy. In semi-classical quantum
mechanics, quantisation rules select sequences of KAM tori that correspond to
quantum levels. Hence a global labeling of quantum levels by two quantum
numbers is not possible.Comment: 11 pages, 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
Breakdown of Lindstedt Expansion for Chaotic Maps
In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336] the
validity of Greene's method for determining the critical constant of the
standard map (SM) was questioned on the basis of some numerical findings. Here
we come back to that analysis and we provide an interpretation of the numerical
results by showing that no contradiction is found with respect to Greene's
method. We show that the previous results based on the expansion in Lindstedt
series do correspond to the transition value but for a different map: the
semi-standard map (SSM). Moreover, we study the expansion obtained from the SM
and SSM by suppressing the small divisors. The first case turns out to be
related to Kepler's equation after a proper transformation of variables. In
both cases we give an analytical solution for the radius of convergence, that
represents the singularity in the complex plane closest to the origin. Also
here, the radius of convergence of the SM's analogue turns out to be lower than
the one of the SSM. However, despite the absence of small denominators these
two radii are lower than the ones of the true maps for golden mean winding
numbers. Finally, the analyticity domain and, in particular, the critical
constant for the two maps without small divisors are studied analytically and
numerically. The analyticity domain appears to be an perfect circle for the SSM
analogue, while it is stretched along the real axis for the SM analogue
yielding a critical constant that is larger than its radius of convergence.Comment: 12 pages, 3 figure
Vanishing Twist near Focus-Focus Points
We show that near a focus-focus point in a Liouville integrable Hamiltonian
system with two degrees of freedom lines of locally constant rotation number in
the image of the energy-momentum map are spirals determined by the eigenvalue
of the equilibrium. From this representation of the rotation number we derive
that the twist condition for the isoenergetic KAM condition vanishes on a curve
in the image of the energy-momentum map that is transversal to the line of
constant energy. In contrast to this we also show that the frequency map is
non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page
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