Coupling schemes for cluster synchronization in coupled Josephson equations (Q1888071)

The paper considers the system of coupled Josephson equations \[ \Phi'' + \alpha \Phi + \beta A \Phi + f(\Phi) =I, \] where \(\Phi=(\phi_1,\phi_2,\dots,\phi_n)^T\), the damping coefficient \(\alpha>0\), the constant input \(I=(I_1,I_2,\dots,\) \(I_n)^T \in \mathbb{R}^n\), \(f(\Phi) =(\sin(\phi_1),\sin(\phi_2),\dots,\sin(\phi_n))^T\). \(\beta>0\) measures the coupling strength and \(A\) is a real matrix reflecting the coupling topology. The authors propose an approach for constructing such coupling schemes, i.e., the matrix \(A\), that a selected cluster synchronization pattern is stable. In particular, a coupling scheme is given to create a synchronization with the frequency ratio \(m_1:m_2:\cdots:m_n\).