Invariant submanifold for series arrays of Josephson junctions

TL;DR

This paper investigates the dynamics of large series arrays of identical Josephson junctions, revealing a reduced invariant submanifold that simplifies understanding their complex behavior.

Contribution

The study extends previous work by identifying a lower-dimensional invariant submanifold using the Ott-Antonsen ansatz, providing new insights into the array's dynamics.

Findings

Reduction of system dimensionality to a one less than previous models

Derivation of flow equations for specific load types

Improved understanding of array synchronization behavior

Abstract

We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of (N - 3) constants of motion, each choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott-Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loading and another with resistive-inductive-capacitive loading. Our results recover (and in some instances improve) earlier findings based on linearization arguments.