Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

Abstract

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines

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In this paper, we give an upper bound (for n ≥ 3) and the least upper bound (for n = 1, 2) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree n, respectively. The results improve the conclusions in [19].

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“…We know that many scholars studied bifurcations of limit cycles of systems with one or more switching lines (see [17,18]). Furthermore, motivated by [1,19], in this paper, we study the maximal number of limit cycles bifurcating from the period annuli of (S 1 ) when they are perturbed by piecewise polynomials of degree n with two switching lines and obtain some new conclusions, which improve the results in [19].…”

“…By perturbing a period annulus with polynomials we can obtain limit cycles. Yang [19] studied the upper bound of the number of limit cycles bifurcated from the period annuli of system (S 1 ), which is perturbed by all piecewise polynomials of degree n with two switching lines. In [12], Liang et al gave a conjecture about the maximal number of limit cycles bifurcated from the period annulus of a linear PWS system with a homoclinic loop, and then Chen and Han [1] confirmed the conjecture.…”

“…The author of [19] took linear transformation of variables x 1 = 1+2y, y 1 = √ 2x for (S 1 ). Then (S 1 ) can be transformed to the following form (omitting the subscript 1)…”

“…For ε = 0, system (1.3) has two centers (1, 0) and (−1, 0) given by H = 0 and H = −1 respectively. From [19] system (1.3) has a family of periodic orbits near each center for ε = 0, see Fig. 1.…”