Stability of Periodic Solutions in Series Arrays of Josephson Junctions with Internal Capacitance

S. Watanabe, J. W. Swift

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


A mystery surrounds the stability properties of the splay-phase periodic solutions to a series array of N Josephson junction oscillators. Contrary to what one would expect from dynamical systems theory, the splay state appears to be neutrally stable for a wide range of system parameters. It has been explained why the splay state must be neutrally stable when the Stewart-McCumber parameter β (a measure of the junction internal capacitance) is zero. In this paper we complete the explanation of the apparent neutral stability; we show that the splay state is typically hyperbolic - either asymptotically stable or unstable - when β > 0. We conclude that there is only a single unit Floquet multiplier, based on accurate and systematic computations of the Floquet multipliers for β ranging from 0 to 10. However, N - 2 multipliers are extremely close to 1 for β larger than about 1. In addition, two more Floquet multipliers approach 1 as β becomes large. We visualize the global dynamics responsible for these nearly degenerate multipliers, and then estimate them accurately by a multiple time-scale analysis. For N = 4 junctions the analysis also predicts that the system converges toward either the in-phase state, the splay state, or two clusters of two oscillators, depending on the parameters.

Original languageEnglish (US)
Pages (from-to)503-536
Number of pages34
JournalJournal of Nonlinear Science
Issue number6
StatePublished - 1997


  • Bifurcation
  • Breakdown of global foliation
  • Josephson junction arrays
  • Multiple time-scale analysis
  • Nonlinear oscillations
  • Stability of periodic solutions

ASJC Scopus subject areas

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


Dive into the research topics of 'Stability of Periodic Solutions in Series Arrays of Josephson Junctions with Internal Capacitance'. Together they form a unique fingerprint.

Cite this