Structured chaos in a devil's staircase of the Josephson junction
Abstract
The phase dynamics of Josephson junctions (JJs) under external electromagnetic radiation is studied through numerical simulations. Currentvoltage characteristics, Lyapunov exponents, and Poincaré sections are analyzed in detail. It is found that the subharmonic Shapiro steps at certain parameters are separated by structured chaotic windows. By performing a linear regression on the linear part of the data, a fractal dimension of D = 0.868 is obtained, with an uncertainty of ±0.012. The chaotic regions exhibit scaling similarity, and it is shown that the devil's staircase of the system can form a backbone that unifies and explains the highly correlated and structured chaotic behavior. These features suggest a system possessing multiple complete devil's staircases. The onset of chaos for subharmonic steps occurs through the Feigenbaum period doubling scenario. Universality in the sequence of periodic windows is also demonstrated. Finally, the influence of the radiation and JJ parameters on the structured chaos is investigated, and it is concluded that the structured chaos is a stable formation over a wide range of parameter values.
 Authors:
 BLTP, JINR, Dubna, Moscow Region 141980 (Russian Federation)
 Department of Physics, University of South Africa, Science Campus, Private Bag X6, Florida Park 1710 (South Africa)
 (State University), Dolgoprudny, Moscow Region 141700 (Russian Federation)
 Institute for Advanced Studies in Basic Sciences, P.O. Box 451951159, Zanjan (Iran, Islamic Republic of)
 Department of Electrical and Electronic Systems Engineering, Utsunomiya University, 712 Yoto, Utsunomiya 3218585 (Japan)
 Publication Date:
 OSTI Identifier:
 22351138
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 24; Journal Issue: 3; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; COMPUTERIZED SIMULATION; ELECTRIC POTENTIAL; ELECTROMAGNETIC RADIATION; FRACTALS; JOSEPHSON JUNCTIONS; LYAPUNOV METHOD; PERIODICITY
Citation Formats
Shukrinov, Yu. M., Botha, A. E., Email: bothaae@unisa.ac.za, Medvedeva, S. Yu., Moscow Institute of Physics and Technology, Kolahchi, M. R., and Irie, A. Structured chaos in a devil's staircase of the Josephson junction. United States: N. p., 2014.
Web. doi:10.1063/1.4890573.
Shukrinov, Yu. M., Botha, A. E., Email: bothaae@unisa.ac.za, Medvedeva, S. Yu., Moscow Institute of Physics and Technology, Kolahchi, M. R., & Irie, A. Structured chaos in a devil's staircase of the Josephson junction. United States. doi:10.1063/1.4890573.
Shukrinov, Yu. M., Botha, A. E., Email: bothaae@unisa.ac.za, Medvedeva, S. Yu., Moscow Institute of Physics and Technology, Kolahchi, M. R., and Irie, A. Mon .
"Structured chaos in a devil's staircase of the Josephson junction". United States.
doi:10.1063/1.4890573.
@article{osti_22351138,
title = {Structured chaos in a devil's staircase of the Josephson junction},
author = {Shukrinov, Yu. M. and Botha, A. E., Email: bothaae@unisa.ac.za and Medvedeva, S. Yu. and Moscow Institute of Physics and Technology and Kolahchi, M. R. and Irie, A.},
abstractNote = {The phase dynamics of Josephson junctions (JJs) under external electromagnetic radiation is studied through numerical simulations. Currentvoltage characteristics, Lyapunov exponents, and Poincaré sections are analyzed in detail. It is found that the subharmonic Shapiro steps at certain parameters are separated by structured chaotic windows. By performing a linear regression on the linear part of the data, a fractal dimension of D = 0.868 is obtained, with an uncertainty of ±0.012. The chaotic regions exhibit scaling similarity, and it is shown that the devil's staircase of the system can form a backbone that unifies and explains the highly correlated and structured chaotic behavior. These features suggest a system possessing multiple complete devil's staircases. The onset of chaos for subharmonic steps occurs through the Feigenbaum period doubling scenario. Universality in the sequence of periodic windows is also demonstrated. Finally, the influence of the radiation and JJ parameters on the structured chaos is investigated, and it is concluded that the structured chaos is a stable formation over a wide range of parameter values.},
doi = {10.1063/1.4890573},
journal = {Chaos (Woodbury, N. Y.)},
number = 3,
volume = 24,
place = {United States},
year = {Mon Sep 01 00:00:00 EDT 2014},
month = {Mon Sep 01 00:00:00 EDT 2014}
}

By analog computer calculations of the resistively and capacitively shunted Josephson junction model, IV characteristics are measured for several choices of the parameters in the Josephson equation. The points, where hysteresis sets in, are related to cubic inflection points in the return map. For different values of the amplitude and the frequency of the imposed ac field the critical line is determined in the (I,G) space, where I is the dc current and G is the damping factor. Furthermore, the subharmonic steps along the critical line form a complete devil's staircase with a fractal dimension Dapprox.0.87 and a decay exponentmore »

Analog simulations of josephson junction in a microwave field. Devil's staircase, fractal dimension, and decay constants
The RSJ model of the Josephson junction in the presence of a microwave field is studied using an analog computer, with special attention to the behavior of this system near or at the critical line, where the set of substeps forms a complete devil's staircase on the IV characteristic. A value of fractal dimension D = 0.868 +/ 0.002 is determined from 240 substeps between the winding numbers W = 0 and W = 1. Four values of decay constants are determined. The results agree very well with the prediction obtained from the onedimensional circle map. A selfsimilarity graph ismore » 
Fractal dimension and selfsimilarity of the devil's staircase in a Josephsonjunction simulator
A Comment on the Letter by M. Hogh Jensen, Per Bak, and Tomas Bohr, Phys. Rev. Lett. 50, 1637 (1983). 
Microwaveinduced ''Devil's Staircase'' structure and ''Chaotic'' behavior in currentfed Josephson junctions
We have obtained the various types of IV characteristics measured experimentally and in analog simulations, by merely changing the junction and the microwave parameters within the same resistively shunted junction model with purely sinusoidal currentphase relation. It was found that the subharmonic steps do exist in the limit b/sub c/..>..0, though they can have finite rounding without thermal noise. The statistical properties of the ''chaotic'' solutions wer e discussed and their effective temperature was defined and calculated. 
Evidence for a devil's staircase in holmium produced by an applied magnetic field
The magnetic structure of holmium has been studied using neutron diffraction when a magnetic field is applied along the {ital c} axis. The field has the effect of suppressing the onset of the commensurate cone phase found at low temperatures in zero field, and instead produces a series of spinslip structures. In contrast to the zerofield diffraction experiments, where a continuous variation of the magnetic wave vector {bold q} was observed, we find that below {approx}15 K the wave vector {bold q} is always commensurate and forms a devil's staircase with increasing field.