Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

Abstract

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.2010 Mathematics Subject Classification. Primary 34C07, 37G15, 34A36.

On the number of limit cycles for Bogdanov-Takens system under perturbations of piecewise smooth polynomials

“…For the piecewise smooth differential systems with the switching lines, there have been a lot of results for example [4,16,18,23,26]. Nowadays many scholars begin to pay attention to the study of piecewise smooth differential systems with the nonlinear switching curves (see [2,6,19,21,22,24,[27][28][29]).…”

On the number of limit cycles bifurcating from the linear center with an algebraic switching curve

“…We are mainly interested in studying the existence of limit cycles for piecewise linear differential systems with two pieces separated by a nonlinear switching curve. In [1], the authors considered the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. They studied the class of discontinuous piecewise linear differential systems obtained by perturbing up to order 2 in the small parameter ǫ the linear center ẋ = y, ẏ = −x, and they obtained that 7 is a lower bound for the Hilbert number of this family.…”

Some open problems in low dimensional dynamical systems

“…Abel differential equations appear in the study of some planar vector fields, see for instance Section 2.1, but they are also interesting by themselves. In general, they write as (11) dx…”

“…It is remarkable that while when A 3 = 0 (the Riccati differential equation) the maximum number of limit cycles is two, there is no upper bound for the number of limit cycles for general Abel differential equations (11), even when the functions A j are trigonometrical polynomials, see [103].…”

“…Two useful results to obtain upper bounds on the number of limit cycles for Abel differential equation (11) are:…”