Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.Comment: 24 pages, 7 figure

Symmetric periodic orbits for the collinear charged 3-body problem

Agraïments: The first author is partially supported by a CAPES grant number 88881.030454/2013-01 from the program CSF-PVE. The second author is supported by grant#2012/10 26 7000 803, Goiás Research Foundation (FAPEG), PRO-CAD/CAPES grant 88881.0 68462/2014-01 and by CNPq-Brazil.In this paper we study the existence of periodic symmetric orbits of the 3-body problem when each body possess mass and an electric charge. The main technique applied in this study is the continuation method of Poincar\'e

Detecting periodic orbits in some 3d chaotic quadratic polynomial differential systems

Using the averaging theory we study the periodic solutions and their linear stability of the 3-dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one-parameter

Detecting periodic orbits in some 3d chaotic quadratic polynomial differential systems

Using the averaging theory we study the periodic solutions and their linear stability of the 3-dimensional chaotic quadratic polynomial differential systems without equilibria studied in [3]. All these differential systems depend only on one-parameter