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A method for estimating the characteristics of the delayed coupling between the oscillatory systems, which is based on the empirical simulation of the phase dynamics with allowance for the phase-noise correlatedness, is proposed. The method efficiency is illustrated FOR the standard stochastic and chaotic oscillators in numerical experiments. Using this method for analyzing climatic time series, we confirm the presence of the delayed influence of the El Nino Southern oscillation on the North Atlantic Oscillation.

Based on a developed and experimentally tested model, self-oscillatory lasing regimes have been studied for a Class A four-frequency gas ring laser with linear coupling of elliptically polarized counter-propagating waves. In the self-oscillatory regime, switching of the intensities, polarization states, and phase differences of the counter-propagating waves is observed, and also a polarization-phase dynamics effect involving displacement of the intensity switching point (with respect to detuning), due to the phase shift as a result of an ellipticity different from zero.

Spiral turbulence observed in Couette-Taylor system has been characterized using the phase diffusion equation suggested by Hegseth et al. [Phys. Rev. Lett. 62, 257 (1989)]. From space-time diagrams, we have measured the diffusion coefficient, the diffusion velocity, and the turbulent spiral pitch.

We report measurements on the phase dynamics of a detuned single mode laser which has in the past shown correspondence to the complex Lorenz equation. The corresponding ranges of periodic pulsing, of detuned chaos and of windows within the chaotic domain are investigated.

Based on the semiclassical density-matrix equations, ultrafast probe gain and phase dynamics in bulk semiconductor optical amplifiers excited by a 200 fs Gaussian pump pulse centered at 1,558 nm are numerically investigated and compared over a wide material gain region (1,460–1,650 nm). It is predicted that the recovery time of phase may be longer or shorter than that of gain depending on probe wavelengths. Besides, as functions of probe wavelength, the dynamic gain range and the dynamic phase range have different shapes. Moreover, the wavelength dependence of phase recovery overshoot is illustrated. At last, the probe chirp dynamics and the wavelength dependence of the chirp characteristics are described. These interesting features result from a restriction of the Kramers–Kronig relations and different ways of the competition or cooperation among the effects of spectral-hole burning, carrier heating and two-photon absorption on subpicosecond timescale over the whole concerned band.

We investigate the relaxation effects on the dynamics of two-component dilute gas Bose-Einstein condensates (BEC) with relatively different two-body interactions and Josephson couplings between the two components. Three types of relaxation effects, i.e., one- and three-body losses and a pure phase relaxation caused by elastic two-body collision between condensed and noncondensed atoms, are examined on the dynamical behavior of a macroscopic superposition, i.e., Schrodinger cat state, of two states with atom-number differences between the two components, which is known to be created by the time evolution in certain parameter regimes. Although three-body losses show a relatively large suppression of the revival behavior of Schrodinger cat state and the Pegg-Barnett phase-difference distribution between the two components for a small-size Schrodinger cat state, one- and three-body loss effects are not shown to directly depend on the size of Schrodinger cat state. In contrast, the pure-phase relaxation effects, causing a reduction of phase-difference distribution and then decaying the Schrodinger cat state, significantly increase with the increase of the size of Schrodinger cat state. These features suggest that a detection of damped collapse-revival behavior is highly possible for medium-size Schrodinger cat states in small-size two-component BECs.

We investigate the Poincare transformation dynamics describing the stochastic motion of a relativistic electron in the regular electrostatic field of a wave packet. The conditions of local instability of the phase trajectories are determined: it is shown that the system is a structurally stable Y system with an attracting set of the hyperbolic type. The relationship between the dynamics at the strange attractor characterized by Lyapunov indices and its statistical structure described by the Kolmogorov entropy is discussed. The fractal dimension of the phase space of the system is determined. The results of a numerical simulation of the effect are presented and discussed.

Analytical and numerical methods were used to study the dynamics of the mapping that describes a determinate dynamic system consisting of a relativistic electron and a wave packet of potential waves. The mapping dynamics is shown to depend essentially on a parameter that characterizes the wave-packet intensity. The system behaves in different ways, depending on that parameter: it executes ergodic motion with displacement or motion with simultaneous regular and irregular trajectories. This mapping is shown to be typical and structurally stable.

Recently there has been a surge of new interest in three-dimensional wake patterns. In the present work, we have devised a method to control the spanwise end conditions and wake patterns using “end suction”, which is both continuously-variable and admits transient control. Classical steady-state patterns, such as parallel or oblique shedding or the “chevron” patterns are simply induced. The wake, at a given Reynolds number, is receptive to a continuous range of oblique shedding angles (θ), rather than to discrete angles, and there is excellent agreement with the “cos θ” formula for oblique-shedding frequencies. We show that the laminar shedding regime exists up to Reynolds numbers (Re) of 205, and that the immense disparity among reported critical Re for wake transition (Re = 140–190) can be explained in terms of spanwise end contamination. Our transient experiments have resulted in the discovery of new phenomena such as “phase shocks” and “phase expansions”, which can be explained in terms of a simple model assuming constant normal wavelength of the wake pattern. Peter Monkewitz (Lausanne) also predicts such transient phenomena from a Guinzburg-Landau model for the wake.

The temperature-dependent helicity modulus Γ(T) of site-diluted triangular arrays of proximity-effect coupled Pb/Cu/Pb Josephson junctions has been extracted from measurements of the complex ac sheet conductance performed in the frequency range 0.1 Hz to 20 kHz in zero magnetic field. At low frequencies Γ(T) exhibits the characteristic superfluid drop predicted by the Kosterlitz-Thouless theory, demonstrating that a disordered array behaves like a homogeneous two-dimensional system at length scales-larger than the percolation correlation length.

Starting from the microscopic BCS-interaction Hamiltonians inside and between the superconductors of a tunnel junction sandwich, the macroscopic part of the Schrodinger dynamics on the set of pure physical phase states is derived with the methods of algebraical statistical mechanics. The resulting nonlinear differential equations for the physical parameters, like the macroscopic phase difference, are solved numerically and compared with the Josephson relations. It is shown that the frequency of the tunnel current has sharp peaks, if it is considered as a function of applied voltage and phase difference. The calculated peak-parameters align to the characteristic fork of bifurcating systems in the voltage-phase parameterspace.

The time variation of the quantum phase difference and the radiated power of two resistively shunted junctions in a superconducting loop are analyzed using a first harmonic approximation method. It is shown explicitly that an output coupling impedance connected across the junctions plays an important role in determining the output power as well as the cutoff frequency in the radiation spectrum. The radiated power of an array of dc SQUIDs is compared with the power output of a synchronized series array in a resonant cavity. WithN superconducting weak links coupled to a fixed load, the SQUID array suffers an output power saturation for largeN; our analysis shows that it is very difficult to obtain radiated power higher than a few microwatts. On the other hand, a cavity-coupled large array gives an output power proportional toN with a narrow linewidth, suggesting that it could be very useful in practice.

Giant fractional Shapiro steps have been observed in Josephson junction arrays as resulting from magnetic flux quantization in the two-dimensional array. We demonstrate experimentally the appearance of giant fractional Shapiro steps in anisotropic Josephson junction arrays as unambiguous evidence of a skewed current phase relationship. Introducing anisotropy in the array results in a giant collective high frequency response that reflects the properties of a single junction, as evidenced by the observation of a Fraunhofer like magnetic field dependence of the total critical current of the system. The observed phase dynamics can be perfectly captured within an extended resistively shunted Josephson junction model. These results directly indicate the potential of Josephson junction arrays to explore the current phase relation in a very broad frequency range (down to 50 MHz) and in a wide variety of novel link materials exhibiting non-conventional current phase relationships. Giant fractional Shapiro steps have been observed in Josephson junction arrays as resulting from magnetic flux quantization in the two-dimensional array. Here, the authors demonstrate the observation of giant fractional Shapiro steps in an anisotropic Josephson junction array, as unambiguous evidence of a skewed current phase relationship.

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.

Taking into account some philosophical notionson realism a reformulation of“macro-realism” according to Leggett-Garg isput forward. A macroscopic phase dynamics based on amicroscopic SQUID-model is discussed in this context.

The effects of the amplitude nonuniformity of the magnetic field in a betatron on the forced and betatron radial oscillations of electrons during extraction (or discharge toward the target) at the end of the acceleration cycle are examined. The amplitude and phase dynamics of radial betatron oscillations are studied by means of an analog computer.

Stepping frequency is tightly coupled to walking velocity during natural locomotion. In a recent model, we demonstrated that walking velocity determines stride frequency, governs the active feedback control of the swing and determines the swing phase dynamics that governs foot movement. Here, we questioned whether the swing phase dynamics reflect independent effects of stride frequency and walking velocity. Foot movements were measured with a motion detection system (Optotrak) while subjects walked at 0.6-2.1 m/s on a treadmill. Stepping frequencies of 1.3-2.8 Hz were generated with pacing cues at each walking velocity. In the 'iso-velocity' condition, peak forward toe velocity during the swing phases was related to walking velocity and did not vary with alterations in stride frequency. In the 'iso-frequency' condition, in contrast, stepping frequency altered the relationship between toe acceleration and toe position in the fore-aft direction. The cycle frequency, main sequence (peak velocity vs. amplitude) relationships, and the shape of the phase-plane trajectories of the swing phases also reflected this relationship. The data were modeled by decoupling stepping frequency from walking velocity, while maintaining active feedback control dependent on frequency. The latter predicted both the dominant shape of the phase plane trajectories and the main sequence relationships. Thus, according to the model, walking velocity and stride frequency are independent central variables that control the dynamics of the swing phases and stepping. The ability to decouple stride frequency from walking velocity may help in navigating over uneven terrain or when executing curved trajectories while maintaining a constant velocity.

During the last several years, the HL-2A experiment has made significant progress in the following areas: (1) lower-hybrid wave (LHW) heating and current drive, (2) plasma confinement and turbulent transport, (3) magnetohydrodynamic (MHD) instabilities and energetic particle physics and (4) H-mode and edge localized mode (ELM) control. The results show that the LHW system working in the co-current mode can reach higher driving efficiency and full non-inductive lower-hybrid current drive (LHCD) has been achieved. The intrinsic poloidal torque characterized by the divergence of the residual stress is deduced from synthesis for the first time. The dynamics of spectral symmetry breaking in drift wave turbulence is in good agreement with the development of the poloidal torque to drive the edge poloidal flow. The influence of the cross-phase dynamics on turbulent stress was also investigated. The ion internal transport barrier has been observed in the NBI-heated plasma, and inside the barrier the ion thermal transport is reduced to the neoclassical level. Besides, micro-turbulence is modulated by the rotation frequency of the magnetic island, and this modulation effect is related to a critical island width. Strong E × B shear is found at the island boundary. Three kinds of axisymmetric modes, beta-induced Alfven eigenmode (BAE), toroidal Alfven eigenmode (TAE) and the ellipticity-induced Alfven eigenmode (EAE), are found to be driven unstable by nonlinear mode coupling between Alfven eigenmodes and tearing mode which is well explained by the nonlinear gyrokinetic theory. The fishbone and tearing modes were actively controlled by the electron cyclotron resonance heating (ECRH). The dynamics of the edge plasma flows and turbulence during the L–I–H transition have been dedicatedly investigated. The geodesic acoustic mode (GAM) and limit cycle oscillation (LCO) coexist for a short time and disappear in the H-mode plasma with the increasing of E × B shear flow before the I–H transition, which plays an important role in the turbulence suppression. Different techniques, such as LHW, ECRH, resonant magnetic perturbation (RMP), and impurity seeding by the laser blow-off (LBO) and supersonic molecular beam injection (SMBI), have been successfully applied to control the large ELMs. It has been found that pedestal turbulence enhancement might be responsible for the observed mitigation effect.

Random lasers (RLs) are remarkable experimental platforms to advance the understanding of complex systems phenomena, such as the replica-symmetry-breaking (RSB) spin glass phase, dynamics modes correlations, and turbulence. Here we study these three phenomena jointly in a Nd:YAG based RL synthesized for the first time using a spray pyrolysis method. We propose a couple of modified Pearson correlation coefficients that are simultaneously sensitive to the emergence and fading out of photonic intermittency turbulent-like effects, dynamics evolution of modes correlations, and onset of RSB behavior. Our results show how intertwined these phenomena are in RLs, and suggest that they might share some common underlying mechanisms, possibly approached in future theoretical models under a unified treatment.

Upon excitation with an intense laser pulse, a symmetry-broken ground state can undergo a non-equilibrium phase transition through pathways different from those in thermal equilibrium. The mechanism underlying these photoinduced phase transitions has long been researched in the study of condensed matter systems1, but many details in this ultrafast, non-adiabatic regime still remain to be clarified. To this end, we investigate the light-induced melting of a unidirectional charge density wave (CDW) in LaTe3. Using a suite of time-resolved probes, we independently track the amplitude and phase dynamics of the CDW. We find that a fast (approximately 1 picosecond) recovery of the CDW amplitude is followed by a slower re-establishment of phase coherence. This longer timescale is dictated by the presence of topological defects: long-range order is inhibited and is only restored when the defects annihilate. Our results provide a framework for understanding other photoinduced phase transitions by identifying the generation of defects as a governing mechanism. Three different ultrafast probes investigate a non-adiabatic phase transition and find substantial evidence of topological defects inhibiting the reformation of the equilibrium phase.

We investigate theoretical aspects of a reduced-order model for transverse galloping, which is derived directly from the full governing equations, i.e., the fluid mass and momentum balances, the fluid far-field boundary condition, the kinematic and dynamic conditions at the interface, and the bluff body equation of motion. We show that the presence of low-intensity turbulence and back-action from the slow transverse oscillations of the bluff body yield a correction to the hydrodynamic force of the quasi-steady theory in the form of additive random excitation. The reduced-order model consists of a pair of nonlinear Langevin equations for the amplitude and phase of the transverse motion of the bluff body. We show that while the phase dynamics is associated with a strongly diffusive random walk motion, the amplitude dynamics is associated with a relatively weak diffusion and can be mapped onto the motion of an overdamped particle trapped in a potential well. This mapping provides a highly useful tool for understanding both the deterministic (no random excitation) and the stochastic (weak random excitation) amplitude dynamics, and hence, for yielding theoretical insights on the overall system dynamics.

Within the framework of the physicomathematical model of evolution of a polydisperse condensate, numerical investigations of the kinetics of atmospheric aerosol droplets in a supersonic two-phase flow past a flat plate were carried out. The gas flow was described by the Reynolds equations with the use of the two-parameter turbulence model. In view of the smallness of the condensate mass fraction in the incoming flow, the inverse effect of the dispersed phase on the gas was not considered. For various regimes of exposure to a flow, the characteristic features of the spatial distribution of the main parameters of the condensate fractions have been studied: the number densities, radii, temperatures, and averaged velocities of microdrops. The dependence of the dispersed phase dynamics on the Mach number and the incoming flow angle of attack has been investigated and the influence of the allowance for the processes of coagulation/fragmentation on the mass spectrum of droplets is shown.

Interface modeling involves surface tension effects and topological transitions such as breakup and coalescence. In order to simulate the dynamics of interfaces as well as phase transitions, we introduce diffuse-interface modeling. This model describes the interface evolution with a high-order diffusion equation based on the van der Waals–Cahn–Hilliard theory of interfaces. Thus, unlike other methods proposed in graphics, phase dynamics is derived from interface physics regardless of the numerical solution. Small-scale interfacial events and phase transitions develop naturally as a result of the Cahn–Hilliard equation. With the fast, stable and conservative spectral method we used and also due to the characteristics of the diffuse-interface description, the numerical method does not suffer from oscillations or other difficulties caused by interface singularities, which could be very problematic in other interface methods, and numerical diffusion and mass loss are minimized. Besides, the implementation is simple and practical. This model also allows the control of interface features, such as thickness and mobility. The Cahn–Hilliard equation can be coupled with the Navier–Stokes equations to model the surface tension force in fluid equations at the macroscopic level. More importantly, the interface dynamics that uses a free-energy function provides the flexibility and versatility of modeling complex material behaviors simply by designing an appropriate free-energy functional. In this way, we modeled certain elastic behaviors and multicomponent phase evolutions, which could be computationally complex in other methods.

We characterized the three-dimensional kinematics and dynamics of quadrupedal gait of young adult rhesus and cynomolgus monkeys while they walked with diagonal and lateral gaits at 0.4-1.0 m/s on a treadmill. Rigid bodies on the wrist, ankle, and back were monitored by an optical motion detection system (Optotrak). Kinematic data could be normalized using characteristic stride length, reducing variance due to different gait styles, to emphasize common characteristics of swing and stance parameters among animals. Mean swing phase durations fell as walking speed increased, but the swing phase durations increased at each walking velocity as a linear function of increases in amplitude, thereby following a main sequence relationship. The phase plane trajectories of the swing phases, i.e., plots of the relation of the rising and falling limb velocity to limb position in the sagittal (X-Z) plane, had unique dynamic characteristics. Trajectories were separable at each walking velocity and increases in swing amplitude were linearly related to peak swing velocities, thus comprising main sequences. We infer that the swing phase dynamics are set by central neural mechanisms at the onset of the swing phases according to the intended amplitude, which in turn is based on the walking velocity in the stance phases. From the many dynamic similarities between swing phases and rapid eye movements, we further suggest that the swing phases may be generated by neural mechanisms similar to those that produce saccades and quick phases of nystagmus from a signal related to sensed or desired walking velocity.

We demonstrate, both from a theoretical and an experimental point of view, the possibility of realizing a weak coupling between two Bose-Einstein condensates trapped in di erent Zeeman states. The weak coupling drives macroscopic quantum oscillations between the condensate populations and the observed current-phase dynamics is described by generalized Josephson equations. In order to highlight the superiuid nature of the oscillations, we investigate the response of a 87 Rb non-condensate (thermal) gas in the same conditions, showing that the thermal oscillations damp more quickly than those of the condensate.

Coupled dynamical systems that operate near the onset of a bifurcation can lead, under certain conditions, to strong signal amplification effects. Over the past years we have studied this generic feature on a wide range of systems, including: magnetic and electric fields sensors, gyroscopic devices, and arrays of loops of superconducting quantum interference devices, also known as SQUIDs. In this work, we consider an array of SQUID loops connected in series as a case study to derive asymptotic analytical approximations to the exact solutions through perturbation analysis. Two approaches are considered. First, a straightforward expansion in which the non-linear parameter related to the inductance of the DC SQUID is treated as the small perturbation parameter. Second, a more accurate procedure that considers the SQUID phase dynamics as non-uniform motion on a circle. This second procedure is readily extended to the series array and it could serve as a mathematical framework to find approximate solutions to related complex systems with high-dimensionality. To the best of our knowledge, an approximate analytical solutions to an array of SQUIDs has not been reported yet in the literature.

In the case of 2D optical patterns, frustration comes from the interplay between the physical constraints (light-matter interaction) and the geometrical constraints (cavity length and structure). Depending on the dynamical parameters, we are able to single out two distinct behaviors. For small diffusion and close to threshold, the system is forced to fulfill the geometrical constraints giving rise to a phase dynamics of quasicrystals. For larger diffusion, the system fragmentates into spatial domains giving rise to a competition between different patterns. By means of a geometrical argument, we show that the spatial distribution of domains is related to the symmetry imposed by the geometrical constraint and that the domain borders are disinclination defects. These defects being the nucleation centers of spatial domains, they trigger the onset of pattern competition.

Exciting developments in earthquake science have benefited from new observations, improved computational technologies, and improved modeling capabilities. Designing realistic supercomputer simulation models for the complete earthquake generation process is a grand scientific challenge due to the complexity of phenomena and range of scales involved from microscopic to global. The present volume - Part II - incorporates computational environment and algorithms, data assimilation and understanding, model applications and iSERVO. Topics covered range from iSERVO and QuakeSim: implementing the international solid earth research virtual observatory by integrating computational grid and geographical information web services; LURR (Load-Unload Response Ratio) described in six papers involving this promising earthquake forecasting model; pattern informatics and phase dynamics and their applications, which was also a highlight in the Workshop; computational algorithms, including continuum damage models and visualization and analysis of geophysical datasets; evolution of mantle material; the state vector approach; and assimilation of data such as geodetic data, GPS data, and seismicity and laboratory experimental data.

The extended Fukuyama-Lee-Rice theory of phase dynamics in quasi one-dimensional charge density wave systems is worked out for the pinned Frohlich mode in the strong pinning limit and for three spatial dimensions. The ac conductivity δ(ω) is expressed in terms of a self energy function Σ(ω) which is evaluated self consistently within the single sitet-matrix approximation. A case of hard momentum cut-off is shown to be inconsistent for reasons of analyticity breaking. A conserving theory for δ(ω) is worked out for a soft Lorentz cut-off and discussed in detail. Analytical results for the low frequency relaxation mode, for the “second dielectric constant”, for the high frequency pinned mode, and for the conductivity step at the frequency of the longitudinal optical phason are given in the descreened limit of low temperatures. The limit of very strong pinning is shown to be isomorphic to the case of the weakly pinned Frohlich mode ind=3, but with different parameters. Finally, a comparison is made between measurements of Re δ(ω) for two semiconducting charge density wave systems and the theories of both weak and strong pinning. The results are inconclusive with weak pinning slightly in favour over strong pinning.

The active mode-locking process of the multimode laser with an external pump modulation is theoretically investigated in the frequency domain within the framework of the continuous-mode approximation. Intermode interaction and mode-coupling effects, including both AM and FM modulations, are naturally considered in a hierarchical equation of the mode components derived from the multimode Maxwell-Bloch equations. It is reduced to a continuous-mode equation that can be solved analytically in a stationary case, and used to discuss the spectral line shape and the phase dynamics of mode-components as a function of modulation amplitude and detuning of the modulation frequency. We predict a novel oscillation existing below the threshold of the ordinary complete mode-locking: The intensity of the total electric field yields a stable pulse train but its phase varies irregularly in time. This semi-locked state is characterized by a nonlinear chirping, an asymmetric spectrum, and drifting phases of the field mode-components.

Finite difference numerical solutions for rotationally symmetric flows in the rotating Couette system are presented. Axial and radial structures of velocity and pressure of Taylor vortices are elucidated. The Reynolds number dependence of their axial Fourier amplitudes is compared with theory. The wave number selection by ramps realized at constant geometry by slow axial decrease of the driving from super-to subcritical values is determined. The phase dynamics of the response to moving a boundary is compared with theory. The long-time effective phase diffusion constant strongly depends on the axial amplitude profile of the vortex structure.