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

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

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions

Orbital solutions derived from radial velocities and time delays for four {\it Kepler} systems with A/F-type (candidate) hybrid pulsators

The presence of A/F-type {\it Kepler} hybrid stars extending across the entire $\delta$ Sct-$\gamma$ Dor instability strips and beyond remains largely unexplained. In order to better understand these particular stars, we performed a multi-epoch spectroscopic study of 49 candidate A/F-type hybrid stars and one cool(er) hybrid object detected by the {\it Kepler} mission. We determined a lower limit of 27 % for the multiplicity fraction. For six spectroscopic systems, we also reported long-term variations of the time delays. For four systems, the time delay variations are fully coherent with those of the radial velocities and can be attributed to orbital motion. We aim to improve the orbital solutions for those systems with long orbital periods (order of 4-6 years) among the {\it Kepler} hybrid stars. The orbits are computed based on a simultaneous modelling of the RVs obtained with high-resolution spectrographs and the photometric time delays derived from time-dependent frequency analyses of the {\it Kepler} light curves. We refined the orbital solutions of four spectroscopic systems with A/F-type {\it Kepler} hybrid component stars: KIC 4480321, 5219533, 8975515 and KIC 9775454. Simultaneous modelling of both data types analysed together enabled us to improve the orbital solutions, obtain more robust and accurate information on the mass ratio, and identify the component with the short-period $\delta$ Sct-type pulsations. In several cases, we were also able to derive new constraints for the minimum component masses. From a search for regular frequency patterns in the high-frequency regime of the Fourier transforms of each system, we found no evidence of tidal splitting among the triple systems with close (inner) companions. However, some systems exhibit frequency spacings which can be explained by the mechanism of rotational splitting.Comment: 11 pages, 15 figures and 5 tables. Accepted for publication in A&