Center-Focus Problem for Discontinuous Planar Differential Equations

Abstract

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls.

Limit cycles in planar piecewise linear differential systems with nonregular separation line

“…We extend to Σ α -piecewise linear vector fields the presentation given in [5] for the Σ π case. The method described uses the decomposition of a one-form given in [11,12] but for higher order perturbations, in polar coordinates, as was introduced in [15].…”

“…More specifically, the method used in this work, see [15], is a generalization to piecewise linear systems of the extension to higher order perturbations, see [21], of the method of Françoise. The main application in [15] is the computation of the Lyapunov constants for piecewise systems and their use in the center-focus problem. Other applications of this method can be found in [30,31].…”

“…The method described in [15] is based on a decomposition of certain one-forms associated to the expression of the vector field in polar coordinates. The decomposition, see Section 2, is done in such a way that it simplifies the computations of the first nonzero term, M N (ρ), of the expansion in ε of the return map associated to the vector field defined on the positive x-axis, so that…”

Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus

“…It is reminiscent of the decompositions used by Françoise [6,7,8] and it is a generalization of Lemma 2.3 of Gasull and Torregrosa given in [11]. This generalization has been also described in [16] and [17].…”

“…Finally we recall some related works, where the computation of the Poincaré first return map is considered, are Gasull and Torregrosa [10,11], Iliev [16], Iliev and Perko [17], Poggiale [22] and Roussarie [25]. The results of these papers allow to compute the first non null term of the Poincaré map, F n (r) = r n (2π), just under the assumption that F i (r) = r i (2π) ≡ 0, for i = 1, .…”