Dra. Garcia-Saldaña, Johanna

In this paper, we apply the Lyapunov center theorem, Weinstein–Moser theorem and the averaging theory of second order to prove the existence of periodic orbits of a one-parameter generalized Hénon–Heiles Hamiltonian system which includes the classical one. We show that this system has at least two-families of stable periodic orbits for energy level h  >  0.

The notion of center goes back to Poincar´e and Dulac, see [10, 6]. They defined a center for a vector field on the real plane as a singular point having a neighborhood filled of periodic orbits with the exception of the singular point. The problem of distinguishing when a monodromic singular point is a focus or a center, known as the focus-center problem started precisely with Poincar´e and Dulac and is still active nowadays with many questions still unsolved. These last years also the centers are perturbed for studying the limit cycles bifurcating from their periodic solutions, see for instance. We recall that a global center for a vector field on the plane is a singular point p having R 2 filled of periodic orbits with the exception of the singular point. The easiest global center is the linear center ˙x = −y, ˙y = x. It is known (see [11, 2]) that quadratic polynomial differential systems have no global centers. The global degenerated centers of homogeneous or quasihomogeneous polynomial differential systems were characterized in [4] and [8], respectively. However the characterization of the global centers in the cases that the center is nilpotent or of linear-type has not been done. This is the first paper in which such classification is done for the linear-type centers for the systems having a linear part at the origin with purely imaginary eigenvalues and cubic homogeneous nonlinearities.

One of the classical and difficult problems in the qualitative theory of differential systems in the plane is the characterization of their centers. In this paper we characterize the linear and nilpotent global centers of polynomial differential systems with quintic homogeneous terms, with the symmetry (x,y,t) → (−x,y,−t) and without infinite singular points.

The Milky Way is a gigantic spiral-shaped disk with a bright, central bulge containing over 100 billion stars that revolves around the central core, the galaxy moves continuously, likewise, our solar system is also in motion. The solar system is located about 3/4 of the way out from the center in one of the galaxy’s spiral arms. Since the first images of the Milky Way as individual stars obtained by Galileo Galilei, the knowledge of this galaxy has increasingly grown, but even now, is very far from being complete. Many scholars have directed their attention to its study, making it an extensive area of research by using all the available tools existing within their area of expertise. Galaxies are defined as large groups of stars together with dust and gas that are held together by the action of the gravity force. Since galaxies are very complex dynamical objects, it becomes important to study their dynamics. One way to do so is to consider it under the realm of the n-body problem for large n, integrating the equations of motion for systems up to thousands of stars by making use of extensive numerical computations [1], or carry out studies from a theoretical point of view.

In this paper, we characterize the global nilpotent centers of polynomial differential systems of the linear form plus cubic homogeneous terms.

We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3-dimensional polynomial system. We illustrate these points of view with several examples. One of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions. As a first application we use these new approaches to prove that all star-like reversible limit cycles are algebraic. As a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star-like reversible center. As far as we know this is the f irst time that this question is solved in full generality. Somehow, this function plays a similar role that an Abelian integral for studying perturbations of Hamiltonian systems.