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

S. Watanabe, James W Swift

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

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
Volume7
Issue number6
StatePublished - Nov 1997

Fingerprint

Floquet multipliers
Josephson Junction
multipliers
Capacitance
Josephson junctions
Periodic Solution
capacitance
Internal
Multiplier
Series
System theory
Multiple Time Scales
Dynamical systems
Global Dynamics
Systems Theory
Asymptotically Stable
oscillators
Dynamical system
Unstable
Converge

Keywords

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

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Stability of Periodic Solutions in Series Arrays of Josephson Junctions with Internal Capacitance. / Watanabe, S.; Swift, James W.

In: Journal of Nonlinear Science, Vol. 7, No. 6, 11.1997, p. 503-536.

Research output: Contribution to journalArticle

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