Chaotic dynamics from coupled magnetic monodomain and Josephson current.

The ordinary (superconductor-insulator-superconductor) Josephson junction cannot exhibit chaos in the absence of an external ac drive, whereas in the superconductor-ferromagnet-superconductor Josephson junction, known as the φ_{0} junction, the magnetic layer effectively provides two extra degrees of freedom that can facilitate chaotic dynamics in the resulting four-dimensional autonomous system. In this work, we use the Landau-Lifshitz-Gilbert model for the magnetic moment of the ferromagnetic weak link, while the Josephson junction is described by the resistively capacitively shunted-junction model. We study the chaotic dynamics of the system for parameters surrounding the ferromagnetic resonance region, i.e., for which the Josephson frequency is reasonably close to the ferromagnetic frequency. We show that, due to the conservation of magnetic moment magnitude, two of the numerically computed full spectrum Lyapunov characteristic exponents are trivially zero. One-parameter bifurcation diagrams are used to investigate various transitions that occur between quasiperiodic, chaotic, and regular regions as the dc-bias current through the junction, I, is varied. We also compute two-dimensional bifurcation diagrams, which are similar to traditional isospike diagrams, to display the different periodicities and synchronization properties in the I-G parameter space, where G is the ratio between the Josephson energy and the magnetic anisotropy energy. We find that as I is reduced the onset of chaos occurs shortly before the transition to the superconducting state. This onset of chaos is signaled by a rapid rise in supercurrent (I_{S}⟶I) which corresponds, dynamically, to increasing anharmonicity in phase rotations of the junction.

Inertial effects in the dc+ac driven underdamped Frenkel-Kontorova model: Subharmonic steps, chaos, and hysteresis.

The effects of inertial terms on the dynamics of the dc+ac driven Frenkel-Kontorova model were examined. As the mass of particles was varied, the response of the system to the driving forces and appearance of the Shapiro steps were analyzed in detail. Unlike in the overdamped case, the increase of mass led to the appearance of the whole series of subharmonic steps in the staircase of the average velocity as a function of average driving force in any commensurate structure. At certain values of parameters, the subharmonic steps became separated by chaotic windows while the whole structure retained scaling similar to the original staircase. The mass of the particles also determined their sensitivity to the forces governing their dynamics. Depending on their mass, they were found to exhibit three types of dynamics, from dynamical mode-locking with chaotic windows, through to a typical dc response, to essentially a free-particle response. Examination of this dynamics in both the upforce and downforce directions showed that the system may not only exhibit hysteresis, but also that large Shapiro steps may appear in the downforce direction, even in cases for which no dynamical mode-locking occurred in the upforce direction.

Devil's staircase and the absence of chaos in the dc- and ac-driven overdamped Frenkel-Kontorova model.

The devil's staircase structure arising from the complete mode locking of an entirely nonchaotic system, the overdamped dc+ac driven Frenkel-Kontorova model with deformable substrate potential, was observed. Even though no chaos was found, a hierarchical ordering of the Shapiro steps was made possible through the use of a previously introduced continued fraction formula. The absence of chaos, deduced here from Lyapunov exponent analyses, can be attributed to the overdamped character and the Middleton no-passing rule. A comparative analysis of a one-dimensional stack of Josephson junctions confirmed the disappearance of chaos with increasing dissipation. Other common dynamic features were also identified through this comparison. A detailed analysis of the amplitude dependence of the Shapiro steps revealed that only for the case of a purely sinusoidal substrate potential did the relative sizes of the steps follow a Farey sequence. For nonsinusoidal (deformed) potentials, the symmetry of the Stern-Brocot tree, depicting all members of particular Farey sequence, was seen to be increasingly broken, with certain steps being more prominent and their relative sizes not following the Farey rule.

Driven kinks in the anharmonic Frenkel-Kontorova model.

Multiple and supersonic topological excitations (kinks) driven by an external dc force in the Frenkel-Kontorova model (a chain of atoms subjected to a periodic substrate potential) with the exponential interatomic interaction are studied with the help of numerical simulation. The simulation results are interpreted in terms of dynamics of two limiting cases, the exactly integrable sine-Gordon equation and the Toda chain. The stability of driven kinks and scenarios of their destruction are described for a wide range of model parameters.

Breather induced modification of the speed of sound.

The modification of the speed of sound induced by the presence of breathers in a quasi-one-dimensional magnetic chain is examined in the framework of the sine-Gordon model. Within the "pseudoharmonic" phonon approximation it was found that the spin-phonon interaction could have a significant influence on the phonon frequencies, and it consequently leads to the modification of the speed of sound which, in the case of breathers, shows quite different magnetic field and temperature behavior when compared to the case of the linear excitations and solitons. An interesting possibility of an indirect experimental examination of the breathers arises from these predictions.

Kinetic properties of multiquanta Davydov-like solitons in molecular chains.

The Fokker-Planck equation for multivibron solitons interacting with lattice vibrations in a molecular chain has been derived by means of the nonequilibrium statistical operator method. It was shown that a soliton undergoes diffusive motion characterized by two substantially different diffusion coefficients. The first one corresponds to the ordinary (Einsteinian or dissipative) diffusion and characterizes the soliton Brownian motion, while the second one corresponds to the anomalous diffusion connected with frictionless displacement of the soliton center of mass coordinate due to the interaction with phonons. Both processes are the consequence of the Cherenkov-like radiation of phonon quanta arising when soliton velocity approaches the phase speed of sound.