1,404 research outputs found
On a computer-aided approach to the computation of Abelian integrals
An accurate method to compute enclosures of Abelian integrals is developed.
This allows for an accurate description of the phase portraits of planar
polynomial systems that are perturbations of Hamiltonian systems. As an
example, it is applied to the study of bifurcations of limit cycles arising
from a cubic perturbation of an elliptic Hamiltonian of degree four
On the number of limit cycles of the Lienard equation
In this paper, we study a Lienard system of the form dot{x}=y-F(x),
dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a
sequence of algebraic approximations to the equation of each limit cycle of the
system. This sequence seems to converge to the exact equation of each limit
cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd
multiplicity are related to the number and location of the limit cycles of the
system.Comment: 10 pages, 5 figures. Submitted to Physical Review
Localizing limit cycles : from numeric to analytical results
Presentation given by participants of the joint international multidisciplinary workshop MURPHYS-HSFS-2016 (MUltiRate Processes and HYSteresis; Hysteresis and Slow-Fast Systems), which was dedicated to the mathematical theory and applications of multiple scale systems and systems with hysteresis, and held at the Centre de Recerca Matemàtica (CRM) in Barcelona from June 13th to 17th, 2016This note presents the results of [4]. It deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Liénard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov syste
Gravitomagnetic Jets
We present a family of dynamic rotating cylindrically symmetric Ricci-flat
gravitational fields whose geodesic motions have the structure of
gravitomagnetic jets. These correspond to helical motions of free test
particles up and down parallel to the axis of cylindrical symmetry and are
reminiscent of the motion of test charges in a magnetic field. The speed of a
test particle in a gravitomagnetic jet asymptotically approaches the speed of
light. Moreover, numerical evidence suggests that jets are attractors. The
possible implications of our results for the role of gravitomagnetism in the
formation of astrophysical jets are briefly discussed.Comment: 47 pages, 8 figures; v2: minor improvements; v3: paragraph added at
the end of Sec. V and other minor improvements; v4: reference added, typos
corrected, sentence added on p. 24; v5: a few minor improvement
Effects of a localized beam on the dynamics of excitable cavity solitons
We study the dynamical behavior of dissipative solitons in an optical cavity
filled with a Kerr medium when a localized beam is applied on top of the
homogeneous pumping. In particular, we report on the excitability regime that
cavity solitons exhibits which is emergent property since the system is not
locally excitable. The resulting scenario differs in an important way from the
case of a purely homogeneous pump and now two different excitable regimes, both
Class I, are shown. The whole scenario is presented and discussed, showing that
it is organized by three codimension-2 points. Moreover, the localized beam can
be used to control important features, such as the excitable threshold,
improving the possibilities for the experimental observation of this
phenomenon.Comment: 9 Pages, 12 figure
Liouvillian integrability of gravitating static isothermal fluid spheres
Agraïments: FEDER/UNAB10-4E-378.We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, authonomous second order ODE for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent authonomous Lotka-Volterra quadratic polynomial differential system in R2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka-Volterra system possesses Liouvillian first integrals for n = −1, −3, −5, which descend from the existence of invariant algebraic curves of degree one, and for n = −6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka-Volterra system, and consequently of the second order ODE for the mass function must be non Liouvillian. This makes quite unlikely the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric
A search for pulsations in the HgMn star HD 45975 with CoRoT photometry and ground-based spectroscopy
The existence of pulsations in HgMn stars is still being debated. To provide
the first unambiguous observational detection of pulsations in this class of
chemically peculiar objects, the bright star HD 45975 was monitored for nearly
two months by the CoRoT satellite. Independent analyses of the light curve
provides evidence of monoperiodic variations with a frequency of 0.7572 c/d and
a peak-to-peak amplitude of ~2800 ppm. Multisite, ground-based spectroscopic
observations overlapping the CoRoT observations show the star to be a
long-period, single-lined binary. Furthermore, with the notable exception of
mercury, they reveal the same periodicity as in photometry in the line moments
of chemical species exhibiting strong overabundances (e.g., Mn and Y). In
contrast, lines of other elements do not show significant variations. As found
in other HgMn stars, the pattern of variability consists in an absorption bump
moving redwards across the line profiles. We argue that the photometric and
spectroscopic changes are more consistent with an interpretation in terms of
rotational modulation of spots at the stellar surface. In this framework, the
existence of pulsations producing photometric variations above the ~50 ppm
level is unlikely in HD 45975. This provides strong constraints on the
excitation/damping of pulsation modes in this HgMn star.Comment: Accepted for publication in A&A, 14 pages, 15 colour figures (revised
version after language editing
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