Limit cycles of planar discontinuous piecewise linear Hamiltonian systems in three regions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg" display="inline" id="d1e222"> <mml:mi>Y</mml:mi> </mml:math> -type

In recent years there has been a significant interest in studying discontinuous piecewise differential systems, mainly due to the wide range of applications in modeling natural phenomena. To understand the dynamics of these systems in the plane one challenge is to control their number of limit cycles. In this paper we study the existence of limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three zones separated by the line Y = { ( x , y ) : x ≥ 0 and y = 0 } ∪ { ( x , y ) : x = 0 and y ≥ 0 } ∪ { ( x , y ) : x ≤ 0 and y = 0 } . We provide the maximum number of crossing limit cycles intersecting each branch of Y in one point, and intersecting two branches of the Y each one in two points. So we have solved the extension of the 16th Hilbert problem to this class of differential systems. • Piecewise differential systems has been study intensively in these last years due to their applications. • One of the main difficulties in the study of these systems is to control the existence or not of their limit cycles. • Here we characterize the maximum number of limit cycles for 27 classes of piecewise differential systems. • In particular, we have solved the extension of the 16th Hilbert problem for these classes of piecewise differential systems.