Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve

We study how to change the maximum number of limit cycles of the discontinuous piecewise linear differential systems with only two pieces in function of the degree of the discontinuity of the algebraic curve between the two linear differential systems. These discontinuous differential systems appear frequently in applied sciences

Crossing limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines

Limit cycles in Filippov systems having a circle as switching manifold

It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here, we study the limit cycles of the planar discontinuous piecewise linear differential systems separated by a circle when both linear differential systems are centers. Our main results show that such discontinuous piecewise differential systems can have zero, one, two, or three limit cycles, but no more limit cycles than three

Limit Cycles of Polynomially Integrable Piecewise Differential Systems

In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides. We assume that at least one of the systems is Hamiltonian. Under this assumption, piecewise differential systems have no more than one limit cycle. This study characterizes linear differential systems with polynomial first integrals

Crossing limit cycles for a class of piecewise linear differential centers separated by a conic

These last years the study of the version of Hilbert's 16th problem for piecewise linear differential systems in the plane, has increased strongly and there are many papers studying the maximum number of crossing limit cycles when the differential system is defined in two zones separated by a straight line, in particular in [11, 13] it was proved that piecewise linear differential centers separated by a straight line have no crossing limit cycles, but in the papers [14, 15] it was shown that the maximum number of crossing limit cycles of piecewise linear differential centers, can change depending of the shape of the discontinuity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

Acord transformatiu CRUE-CSICWe 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 weak-foci 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

Limit cycles from a monodromic infinity in planar piecewise linear systems

Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier. It is shown that the Hopf bifurcation at infinity can have degeneracies of co-dimension three and, in particular, up to three limit cycles can bifurcate from the periodic orbit at infinity. This provides a new mechanism to explain the claimed maximum number of limit cycles in this family of systems. The centers at infinity classification together with the limit cycles bifurcating from them are also analyzed.Comment: 24 pages, 5 figure