IN THIS PAPER WE STUDY THE CONTINUOUS AND DISCONTINUOUS PLANAR PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY FOUR LINEAR CENTERS SEPARATED BY THREE PARALLEL STRAIGHT LINES DENOTED BY ?={(X,Y)?R2:X=?P,X=0,X=Q, P,Q>0}. WE PROVE THAT WHEN THESE PIECEWISE DIFFERENTIAL SYSTEMS ARE CONTINUOUS THEY HAVE NO LIMIT CYCLES. WHILE FOR THE DISCONTINUOUS CASE WE SHOW THAT THEY CAN HAVE AT MOST FOUR LIMIT CYCLES AND WE ALSO PROVIDE EXAMPLES OF SUCH SYSTEMS WITH ZERO, ONE, AND TWO LIMIT CYCLES. IN PARTICULAR WE HAVE SOLVED THE EXTENSION OF THE 16TH HILBERT PROBLEM TO THIS CLASS OF PIECEWISE DIFFERENTIAL SYSTEMS.

THE STUDY OF LIMIT CYCLES OF PLANAR DIFFERENTIAL SYSTEMS IS ONE OF THE MAIN AND DIFFICULT PROBLEMS FOR UNDERSTANDING THEIR DYNAMICS. THUS THE OBJECTIVE OF THIS PAPER IS TO STUDY THE LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS IN THE PLANE SEPARATED BY A NON-REGULAR LINE ?. MORE PRECISELY, WE SHOW THAT A CLASS OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY AN ARBITRARY QUADRATIC CENTER, AN ARBITRARY LINEAR CENTER AND THE LINEAR CENTER X?=?Y, Y?=X HAVE AT MOST TWO CROSSING LIMIT CYCLES AND WE FIND EXAMPLES OF SUCH SYSTEMS WITH ONE CROSSING LIMIT CYCLE. SO WE HAVE SOLVED THE EXTENSION OF THE 16TH HILBERT PROBLEM TO THIS CLASS OF PIECEWISE DIFFERENTIAL SYSTEMS PROVIDING AN UPPER BOUND FOR ITS MAXIMUM NUMBER OF LIMIT CYCLES.

WE STUDY PLANAR DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY THREE LINEAR HAMILTONIAN SADDLES SEPARATED BY THE NON-REGULAR LINE $ \SIGMA =\{(X,Y)\IN \MATHBB {R}2: (Y=0) \VEE (X=0 \WEDGE Y\GEQ 0)\} $ SIGMA={(X,Y)IS AN ELEMENT OF R2:(Y=0)BOOLEAN OR(X=0 BOOLEAN AND Y >= 0)}. WE PROVE THAT WHEN THE LINEAR HAMILTONIAN SADDLES ARE HOMOGENEOUS THEY HAVE NO LIMIT CYCLES, AND WHEN THEY ARE NON HOMOGENEOUS THEY CAN HAVE AT MOST THREE LIMIT CYCLES HAVING EXACTLY ONE POINT ON EACH BRANCH OF SIGMA, AND AT MOST ONE LIMIT CYCLE HAVING FOUR INTERSECTION POINTS ON SIGMA. MOREOVER WE SHOW THAT THEY CAN HAVE AT MOST ONE LIMIT CYCLE HAVING FOUR INTERSECTION POINTS ON SIGMA AND THREE LIMIT CYCLES HAVING THREE INTERSECTION POINTS ON SIGMA, SIMULTANEOUSLY. THUS WE HAVE SOLVED THE EXTENSION OF THE 16TH HILBERT PROBLEM TO THIS CLASS OF PIECEWISE DIFFERENTIAL SYSTEMS. FURTHERMORE WE SHOW THAT FOR THE THREE TYPES OF COMBINATIONS OF THE LIMIT CYCLES HERE STUDIED IN TWO OF THEM THE UPPER BOUND IS SHARP BY PROVIDING EXAMPLES OF THESE SYSTEMS WITH THE MAXIMUM NUMBER OF POSSIBLE LIMIT CYCLES. FOR THE REMAINING TYPE OF LIMIT CYCLES THE UPPER BOUND IS FOUR AND WE HAVE EXAMPLES WITH THREE LIMIT CYCLES. SO AT THIS MOMENT IT IS AN OPEN PROBLEM TO KNOW IF THE SHARP UPPER BOUND IS EITHER FOUR OR THREE.