Aziza Berbache, Rebeiha Allaoua, Rachid Cheurfa, Ahmed Bendjeddou

Limit Cycles for Piecewise Differential Systems Formed by an Arbitrary Linear System and a Cubic Isochronous Center

In this paper, we study the existence and the maximum number of crossing limit cycles that can exhibit of some class of planar piecewise differential systems formed by two regions and separated by a straight line $x=0 $, where in the left region we define an arbitrary linear differential system and in the right region we define a cubic polynomial differential system with a homogeneous nonlinearity and an isochronous center at the origin. More precisely, we show that these systems may have at most zero or one or two explicit algebraic or non-algebraic limit cycles depending on the type of their linear differential system, i.e., if those systems have foci, center, saddle, node with different eigenvalues, non-diagonalizable node with equal eigenvalues or linear system without equilibrium points.

Mathematics Subject Classification: 34A30, 34C05, 34C25, 34C07, 37G15

Key words and phrases: Discontinuous piecewise differential systems, first integral, Poincaré map, crossing limit cycles