Dynamics in coupled systems of Josephson junctions without capacity effect (Q1299856)

The authors consider the following system of PDEs with homogeneous Neumann and space-periodic boundary conditions \[ \begin{cases} \dot u_1+ \sin u_1+ K(u_1-u_2)- d_1 \Delta u_1= I_1+ b_1\sin \omega t,\\ \dot u_2+\sin u_2+ K(u_2-u_1)- d_2 \Delta u_2= I_2+ b_2\sin \omega t, \end{cases} \tag{1} \] (which may be regarded when the damping \(\alpha\) is large), where \(d_j\), \(K\), \(b_j\), \(I_j\) satisfy natural conditions and \(\Omega\subset \mathbb{R}^n\) is a bounded domain with smooth boundary or \(\Omega= \prod_{j=1}^n (0,L_j)\). The system (1) can be considered as the model of a system of coupled continuous Josephson junctions without capacity effect. The authors prove that in an appropriate functional space, the system (1) has a unqiue one-dimensional invariant continuous periodic curve which is globally attracting and on this curve the restriction of system (1) is exactly related to a system of ODEs, which can be treated as a model of the two-point Josephson junction without capacity effect. Thus the dynamics of (1) is reduced to some one-dimensional system.