Dynamics of Josephson Junction Arrays

The dynamical behavior of Josephson junction arrays, a system of globally coupled nonlinear oscillators, is investigated. Two qualitative cases are considered: the addition of a small signal to the system when it is operating near an instability, and the linear stability of the splay phase dynamical state. Both analytic and numerical techniques are used. We find that the amplification properties of the arrays depend on the type of instability, whether the bifurcation preserves or destroys the coherent state (where all oscillators have the same phase). In the case of the splay phase dynamical state, we find a curious property of neutral stability: if N is the number of Josephson junction oscillators, then the splay phase state is actually a family of periodic orbits foliating an N-2 dimensional invasiant manifold in phase space. Consequently, there are N-2 Lyapunov exponents equal to 0. Further, we find that this neutral stability is broken when a higher order correction (the phase dependent part of the conductance) is included, but not when the arrays are made up of non-identical junctions.