R.D. Parmentier

Future directions of nonlinear dynamics in physical and biological systems

Early in 1990 a scientific committee was formed for the purpose of organizing a high-level scientific meeting on Future Directions of Nonlinear Dynamics in Physical and Biological Systems, in honor of Alwyn Scotts 60th birthday (December 25, 1991). As preparations for the meeting proceeded, they were met with an unusually broad-scale and high level of enthusiasm on the part of the international nonlinear science community, resulting in a participation by 168 scientists from 23 different countries in the conference, which was held July 23 to August 1 1992. The contributions to this present volume have been grouped into the following chapters: (1) Integrability, solitons and coherent structures; (2) Nonlinear evolution equations and diffusive systems; (3) Chaotic and stochastic dynamics; (4) Classical and quantum lattices and fields; (5) Superconductivity and superconducting devices; (6) Nonlinear optics; (7) Davydov solitons and biomolecular dynamics; and (8) Biological systems and Neurophysics.

Self locking of fluxon oscillations in series arrays of niobium Josephson tunnel junctions coupled to a linear resonator

Abstract Spontaneous phase locking of a controlled number from one to ten of all-niobium Josephson fluxon oscillators is obtained at 10 GHz by coupling the oscillator arroy to a linear resonator. The locking range in bias current varies linearly with the number of junctions; the output power, up to 10 nW in a 50 ohm load, varies quadratically. A simple particle-map perturbation theory model efficiently captures the essential experimental phenomenology.

Model studies of long Josephson junction arrays coupled to a high-Q resonator

Series‐biased arrays of long Josephson junction fluxon oscillators can be phase locked by mutual coupling to a high‐Q, linear distributed resonator. A simplified model of such a device, consisting of junctions described by the particle‐map perturbation theory approach which are capacitively coupled to a lumped, linear tank circuit, reproduce the essential experimental observations at a very low computational cost. A more sophisticated model, consisting of partial differential equation descriptions of the junctions, again mutually coupled to a linear tank, substantially confirm the predictions of the simplified model. In the particle‐map model, the locking range in junction bias current increases linearly with the coupling capacitance; in the partial differential equation (p.d.e.) model, this holds up to a certain maximum value of the capacitance, after which a saturation of the locking range is observed. In both models, for a given spread of junction lengths, the existence of a minimum value of the capaci...

KINKS IN THE PRESENCE OF RAPIDLY VARYING PERTURBATIONS

The dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows one to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the small parameter

Threshold analysis for the inverse ac Josephson effect

{\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}1}

Phase locking of fluxon oscillations in long Josephson tunnel junctions with surface losses

, \ensuremath{\omega} being the frequency of the rapidly varying ac driving force. Three physically important examples of such a dynamics, i.e., kinks driven by a direct or parametric ac force, and kinks on a rotating and oscillating background, are analyzed in detail. It is shown that in the main order of the asymptotic procedure the effective equation for the slowly varying field component is a renormalized sine-Gordon equation in the case of the direct driving force or rotating (but phase locked to an external ac force) background, and it is the double sine-Gordon equation for the parametric driving force. The properties of the kinks described by the renormalized nonlinear equations are analyzed, and it is demonstrated analytically and numerically which kinds of physical phenomena may be expected in dealing with the renormalized, rather than the unrenormalized, nonlinear dynamics. In particular, we predict several qualitatively new effects which include, e.g., the perturbation-induced internal oscillations of the 2\ensuremath{\pi} kink in a parametrically driven sine-Gordon model, and the generation of kink motion by a pure ac driving force on a rotating background.

Phase locking of fluxons in spatially inhomogeneous Josephson junctions

Abstract The inverse ac Josephson effect involves rf-induced (Shapiro) steps that cross over the zero-current axis; the phenomenon is of interest in voltage standard applications. The standard analysis of the step height in current, which yields the well-known Bessel-function dependence on an effective ac drive amplitude, is valid only when the drive frequency is large compared with the junction plasma frequency or the drive amplitude is large compared with the zero-voltage Josephson current. Using a first order Krylov-Bogoliubov power-balance approach we derive an expression for the threshold value of the drive amplitude for zero-crossing steps that is not limited to the large frequency or large amplitude region. Comparison with numerical solutions of the RSJ differential equation shows excellent agreement for both fundamental and subharmonic steps. The power-balance value for the threshold converges to the Bessel-function value in the high-frequency limit.

Inverse ac Josephson effect for a fluxon in a long modulated junction

Abstract The influence of surface-resistance dissipation on phase locking of fluxon oscillations in long Josephson junctions of in-line geometry subjected to microwave fields which interact with the fluxon at the junction boundaries is studied by the perturbation- theory map approach and by full integration of the pde model of the junction.

Long Josephson junctions with uniform dc and rf driving currents: General results for the phase‐locked regimes

Abstract We use a simple perturbative approach to study phase locking dynamics of a fluxon in a long Josephson junction in the presence of a periodic spatial inhomogeneity. We derive an analytical experssion for the current size of the phase-locked step as a function of the systems parameters, which is valid in the ultra-relativistic limit. We find that our analysis is in good agreement with direct numerical simulations of the collective-coordinate description of the full system.

On the switching between soliton dynamic states in long Josephson junctions

Abstract We analyze motion of a fluxon in a weakly damped ac-driven long Josephson junction with a periodically modulated maximum Josephson current density. We demonstrate both analytically and numerically that a pure ac bias current can drive the fluxon at a resonant mean velocity determined by the driving frequency and the spatial period of the modulation, provided that the drive amplitude exceeds a certain threshold value. In the range of strongly “relativistic” mean velocities, the agreement between results of a numerical solution of the effective (ODE) fluxon equation of motion and analytical results obtained by means of the harmonic-balance analysis is fairly good; morever, a preliminary PDE result tends to confirm the validity of the collective-coordinate (PDE-ODE) reduction. At nonrelativistic mean velocities, the basin of attraction, in the position-velocity space, for phase-locked solutions becomes progressively smaller as the mean velocity is decreased.