On Piecewise Linear Differential Systems with n Limit Cycles of Arbitrary Multiplicities in Two Zones

In this paper, we demonstrate rich dynamical phenomenon of piecewise linear differential systems having only two zones in the plane. We show that, for any given integer n and any integer tuple \(m=(m_1,m_2,\dots , m_n)\), \( m_i \ge 0 \), for \(i=1,\dots ,n\), there exists an aforementioned system which possesses exactly n limit cycles having multiplicities \(m_1\), \(m_2,\dots , m_n\), respectively. (i.e. there are totally \(m_1+m_2+\cdots +m_n\) limit cycles taking into account of multiplicities). Moreover, we can even choose the separation boundary of the zones to be an algebraic curve.