Limit Cycles Bifurcating of Discontinuous and Continuous Piecewise Differential Systems of Isochronous Cubic Centers with Three Zones

We study the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous cubic centre of discontinuous and continuous piecewise differential systems with three zones formed by the discontinuity set \(\Sigma =\{(x,y)\in \mathbb {R}^2: (y=0) \vee (x=0 \wedge y\ge 0)\}\) $\mathrm{\Sigma }=\{(x,y)\in {\mathbb{R}}^2:(y=0)\vee (x=0\wedge y\ge 0)\}$ . More precisely, we consider the following perturbed systems \(\begin{aligned} \dot{x}=-y+x^2y+ \epsilon p_i(x,y),\ \dot{y}= x+xy^2+ \epsilon q_i(x,y),\ i=1,2,3, \end{aligned}\) $$\begin{array}{c}\hfill \dot{x}=-y+x^{2}y+ϵp_i(x,y),\dot{y}=x+x{y}^2+ϵq_i(x,y),i=1,2,3,\end{array}$$ where \(p_i\) $p_i$ and \(q_i\) $q_i$ are polynomials of degree m. Using the averaging theory of first order, we prove that for \(m=1, 2, 3\) $m=1,2,3$ , at most 3, 9 and 15 limit cycles bifurcate from the periodic annulus of the isochronous cubic centre in the discontinuous case, respectively. While for the continuous case, it can appear 1, 5 and 6 limit cycles from the periodic orbits of these centres, respectively. Furthermore, we extend our study when \(p_i\) $p_i$ and \(q_i\) $q_i$ are homogeneous polynomials with \(1\le m \le 3\) $1\le m\le 3$ , obtaining respectively for \(m=1, 2, 3\) $m=1,2,3$ at most one, seven and at least twelve limit cycles, which bifurcate from the periodic orbits of the isochronous cubic centre.