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Coupling schemes for cluster synchronization in coupled Josephson equations. (English) Zbl 1066.34046
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 $$m_1:m_2:\cdots:m_n$$.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C30 Manifolds of solutions of ODE (MSC2000) 34C60 Qualitative investigation and simulation of ordinary differential equation models
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