**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

**abstract:**

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