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This paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line, Σ, and only one Σ-singularity of monodromic type. The proofs are based in an extension of Rolle's Theorem for dynamical systems on the plane. [ABSTRACT FROM AUTHOR]

We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S 1 , S 2 , S 3 , and S 4 , as well as to non-smooth perturbations of non-smooth centers given by putting different S i 's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper. [ABSTRACT FROM AUTHOR]

Our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is x y = 0. We consider the symmetric case. That is, one vector field defined in the odd quadrants and the other in the even ones. We deal with equilibrium points of center-focus type, with matrices in real Jordan form, in each vector field when the infinity is monodromic. In this case, we provide the center classification at infinity, we prove that the maximum order of a weak focus is five. Moreover, we show the existence of systems exhibiting five limit cycles bifurcating from infinity. [ABSTRACT FROM AUTHOR]

Abstract 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, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center. [ABSTRACT FROM AUTHOR]

A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, ( x ′ , y ′ ) = ( − y + ε f ( x , y , ε ) , x + ε g ( x , y , ε ) ) . In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n , no more than N n − 1 limit cycles appear up to a study of order N . We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Liénard differential systems providing better upper bounds for higher order perturbation in ε , showing also when they are reached. The Poincaré–Pontryagin–Melnikov theory is the main technique used to prove all the results. [ABSTRACT FROM AUTHOR]