The Josephson junction circuit family: Network theory. (English) Zbl 1016.34050
Summary: Here, a qualitative theory for the Josephson junction circuit family is developed using dissipative Hamiltonian systems and the theory of ordinary and stochastic differential equations. Circuit models for the single junction and the flux quantization are presented. Moreover, a novel classification for the Josephson junction circuit family is introduced. Network models for Josephson junction circuits are developed and the physical mechanism for oscillations is explained which is different to the mechanism in semiconductor circuits. It is shown that a general description of periodic solutions in the cylindrical phase space (periodic solution of the second kind) is evident. Several types of hystereses in Josephson junction circuits are explained. For the influence of thermal noise two concepts are presented, the Langevin approach and the application of the Fokker-Planck equation to the Josephson junction. Both are used for the Josephson junction circuit family. Two special kinds of stochastic resonance with practical importance are explained.
The developed theory is applied to several circuits with practical applications and the software package DONANS for the analysis of general nonlinear and noisy systems is presented.
The developed theory is applied to several circuits with practical applications and the software package DONANS for the analysis of general nonlinear and noisy systems is presented.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 82D55 | Statistical mechanics of superconductors |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 94C05 | Analytic circuit theory |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 34C55 | Hysteresis for ordinary differential equations |
Keywords:
Josephson junction circuit family; dissipative Hamiltonian systems; flux quantization; oscillations; semiconductor circuits; periodic solutions; hystereses; thermal noise; Langevin approach; Fokker-Planck equation; stochastic resonanceReferences:
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