### 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 language | English (US) |
---|---|

Pages (from-to) | 503-536 |

Number of pages | 34 |

Journal | Journal of Nonlinear Science |

Volume | 7 |

Issue number | 6 |

State | Published - Nov 1997 |

### Fingerprint

### 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

*Journal of Nonlinear Science*,

*7*(6), 503-536.

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

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 7, no. 6, pp. 503-536.

}

TY - JOUR

T1 - Stability of Periodic Solutions in Series Arrays of Josephson Junctions with Internal Capacitance

AU - Watanabe, S.

AU - Swift, James W

PY - 1997/11

Y1 - 1997/11

N2 - 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.

AB - 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.

KW - Bifurcation

KW - Breakdown of global foliation

KW - Josephson junction arrays

KW - Multiple time-scale analysis

KW - Nonlinear oscillations

KW - Stability of periodic solutions

UR - http://www.scopus.com/inward/record.url?scp=0000984323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000984323&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 503

EP - 536

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 6

ER -