We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by H p (n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then H p (2) ≥ 16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.2010 Mathematics Subject Classification. Primary: 37G15, 37C27. Secundary: 37G10, 34C07. Key words and phrases. Non-smooth differential system, limit cycles in piecewise quadratic differential systems, first and second order perturbations of isochronous quadratic systems, Hilbert number for piecewise quadratic differential systems.

The Bendixson-Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systemsẍ + f ± (x)ẋ + x = 0. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one.

We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system (x , y ) = (−y((x − 1) 2 + y 2 ), x((x − 1) 2 + y 2 )) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. This problem is a generalization of [24] to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree as the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. When the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three types of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario, only 5 limit cycles were found.

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weakfoci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.

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