On the Piecewise Holomorphic Systems with Three Zones

We study piecewise-smooth systems with three zones, $\dot{z} = f_i(z)$, $i = 1,2,3,$ whose discontinuity set $\Sigma$ consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each $f_i:\overline{\mathbb{C}} \to \overline{\mathbb{C}}$ is assumed to be a holomorphic function. We establish conditions ensuring the existence of limit cycles in such systems and provide lower bounds for the maximum number of limit cycle. Our approach combines the Melnikov method, local integrability properties of holomorphic systems, and the existence of normal forms around zeros and poles.