Yuman Böttcher: Structural and Dynamical Failure in RCSJ Networks

Abstract: We consider networks of Josephson junctions described by the Resistively and Capacitively Shunted Junction (RCSJ) model. It describes a dynamical current across every junction and a critical current, which if exceeded by the dynamical current, makes a junction non-superconducting. The dynamical currents are driven by a current applied via a source and a sink. We establish an incidence matrix-based formalism to describe the network. Static and dynamical failures are identified as two distinct failure types in an RCSJ network. Using the established formalism, we find a necessary and sufficient condition for the existence of a fully superconducting state in terms of a maximum flow problem. The max-flow-min-cut theorem and Cheeger inequality are used to connect that condition to the spectral properties of the underlying graph Laplacian. We subsequently derive a sufficient condition expressed in terms of the algebraic connectivity of the network. Dynamical failures are analyzed using the Laplace transform of a linearized small perturbation of the network. For small perturbations, the system is found to be asymptotically stable. The analysis further indicates spectral parameter regimes that may result in cascading failures across the network.