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The study of hyperbolic waves involves various notions which help characterise how these structures evolve. One important facet is the notion of genuine nonlinearity, namely the ability for shocks and rarefactions to form instead of contact discontinuities. In the context of the Whitham modulation equations, this paper demonstrate that a loss of genuine nonlinearity leads to the appearance of a dispersive set of dynamics in the form of the modified Korteweg de-Vries equation governing the evolution of the waves instead. Its form is universal in the sense that its coefficients can be written entirely using linear properties of the underlying waves such as the conservation laws and linear dispersion relation. This insight is applied to two systems of physical interest, one an optical model and the other a stratified hydrodynamics experiment, to demonstrate how it can be used to provide insight into how waves in these systems evolve when genuine nonlinearity is lost.

We investigate nonlinear phase dynamics of an ideal kink mode, induced by E × B flow. Here the phase is the cross phase (θ c) between perturbed stream function of velocity ( ϕ ˜ ) and magnetic field ( ψ ˜ ), i.e. θ c = θ ϕ − θ ψ . A dimensionless parameter, analogous to the Richardson number, R i = 16 γ kink 2 / ω ˆ E 2 (γ kink: the normalized growth rate of the pure kink mode; ω ˆ E : normalized E × B shearing rate) is defined to measure the competition between phase pinning by the current density and phase detuning by the flow shear. When R i > 1, θ c is locked to a fixed value, corresponding to the conventional eigenmode solution. When R i ≤ 1, θ c enters a phase slipping or oscillating state, corresponding to a nonmodal solution. The nonlinear phase dynamics method provides a more intuitive explanation of the complex dynamical behavior of the kink mode in the presence of E × B shear flow.

Driven mean-field threshold systems demonstrate complex observable space-time patterns of behavior that are difficult to understand or predict without knowledge of the underlying dynamics, which are typically unobservable. Here we describe a new method based on phase dynamics techniques to analyze and forecast the space-time patterns of activity in these systems. Application to earthquake data from a typical, seismically active region shows that the method holds considerable promise for forecasting the temporal occurrence of the largest future events. We demonstrate the power of our technique via an application to the difficult problem of earthquake forecasting in southern California.

Transient phase dynamics, synchronization and desynchronization which are stimulus-locked (i.e. tightly time-locked to a repetitively administered stimulus) are studied in two coupled phase oscillators in the presence of noise. The presented method makes it possible to detect such processes in numerical and experimental signals. The time resolution is enormous, since it is only restricted by the sampling rate. Stochastic stimulus locking of the phases or the n:m phase difference at a particular time t relative to stimulus onset is defined by the presence of one or more prominent peaks in the cross-trial distribution of the phases or the n:m phase difference at time t relative to stimulus onset in an ensemble of post-stimulus responses. Unlike the presented approach, both triggered averaging (where an ensemble of post-stimulus responses is simply averaged) and cross-trial cross correlation (CTCC) lead to severe misinterpretations: The oscillators' coupling may cause a transient anti-phase clustering of the post-stimulus responses. Triggered averaging cannot distinguish such a response decorrelation from a mean amplitude decrease of the single responses. CTCC not only depends on the oscillators' phase difference but also on their phases and, thus, inevitably displays "artificial" oscillations that are not related to synchronization or desynchronization.

We discuss the phase dynamics in the system of a set of nonlinear coupled oscillators. We find a direct and simple geometric method to observe the geometric phase for this system, and it provides a possibility for the experimental measurements.

Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two-dimensional geometries, patterns of different symmetries may be simultaneously stable. We study here the phase stability of hexagonal planforms in the framework of amplitude equations and reduced dynamical models close to the instability points.

Hodgkin-Huxley model neurons coupled by weak excitatory interactions are studied by a phase reduction technique. All the information about the coupling between the neurons and their synchronization is then contained in an effective interaction between their phases. One shows analytically that an excitatory coupling can result in an effective inhibition between the neurons reducing their firing rates. Systems of two neurons exhibit bistability and out-of-phase locking. It is suggested that these features may have significant consequences for networks.

Two parallel stacks of coupled Josephson junctions (JJs) are investigated to clarify the physics of transitions between the rotating and oscillating states and their effect on the IV-characteristics of the system. The detailed study of phase dynamics and bias dependence of the superconducting and diffusion currents allows one to explain all features of simulated IV-characteristics and demonstrate the correspondence in their behavior. The coupling between JJ in the stacks leads to the branching of IV-characteristics and a decrease in the hysteretic region. The crucial role of the diffusion current in the formation of the IV-characteristic of the parallel stacks of coupled JJs is demonstrated. We discuss the effect of symmetry in a number of junctions in the stacks and show a decrease of the branching in the symmetrical stacks. The observed effects might be useful for development of superconducting electronic devices based on intrinsic JJs.

The phase dynamics of the coupled Josephson junctions in the framework of CCJJ and CCJJ+DC models is studied. The current voltage characteristics (CVC) are numerically calculated for the stacks with different number of junctions at different model parameters. We manifest the difference of these models for the branching at I = Ic and in the hysteretic region. The essential difference is observed in the breakpoint region, where the longitudinal plasma wave is created. We discuss the main features of both models, related with the role of the diffusion current between the superconducting layers.

The effect of synchronization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in this paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study.

We study the spatial stability of systems at onset of the temporal intermittent transition to chaos. We show that the phase dynamics during laminar episodes, described as orbits near a saddle-node bifurcation of limit cycles, is second order in time and can lead to propagative behaviour.

A study of Hodgkin–Huxley (HH) neuron under external sinusoidal excited stimulus is presented in this paper. As is well known, the stimulus frequency is to be considered as a bifurcate parameter and numerous phenomena, such as synchronization, period and chaos appear alternatively with the changing of the stimulus frequency. For the stimulus frequency less than 2fB (fB being the base frequency in this paper), the simulation results demonstrate that the single HH neuron could completely convey the sinusoidal signal in anti-phase into interspike interval (ISI) sequences. We also report, perhaps for the first time, another kind of phenomenon, the beat phenomenon, which exists in the phase dynamics of the ISI sequences of the HH neuron stimulated by a sinusoidal current. It is shown furthermore that intermittent transition results in the general route to chaos.

Synchrony and antiphase phase locking in synaptically coupled neurons have been studied by employing numerical simulations. By using a Morris–Lecar (ML) model of two coupled stochastic neural oscillators, the effect of noise on the phase dynamics is extensively investigated in two different synaptic-coupling schemes; reciprocal excitatory and excitatory–inhibitory synapses. All possible dynamical features are presented in the parameter space of two important branching parameters, the coupling strength gsyn and the synaptic time delay τd. In this work, antiphase states are observed and analyzed as the breaking dynamics of synchrony. As a transition state from synchrony to antiphase phase locking in the stochastic system, intermittent antiphase states are also reported.

This paper proposes a new feed-forward control strategy for a plant with non-minimum phase dynamics. A Feed-Forward controller is very essential for controlling plants with time-varying reference signals. However, designing this type of controller is a non-trivial problem in case the plant dynamics is non-minimum-phase. This is because, this controller design involves the concept of inversion of the plant model and inverse of a non-minimum-phase plant model is unstable or non-causal. For this problem, usually an approximate inversion of a plant model is applied. An approach with corrected-approximate-inverse (CAI) method is available for feed forward controller design. After analyzing the results and discussion of this CAI technique, it is seen that although it seems to be working perfectly for small duration of time but its performance is unsatisfactory in case of large span of time. Therefore, in this paper, a new feed-forward control strategy has been designed to erase the above said problem. This method has adopted a simple version of an internal plant-model structure in its time domain. The fixed-structure feed forward controller that is constructed using this method is usually a linear amalgamation of a reference trajectory and its time-derivatives with suitable weighting factors. This control strategy has been verified with appropriate simulation results applied to a studied plant and results are compared with that of the CAI technique.

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlev\'e transcendent. In each case the potential can be chosen to be time-independent., Comment: 8 pages, 7 figures. Version 2: stability analysis of the dark solution

Concerning the spontaneous THz electromagnetic wave emission from intrinsic Josephson junctions, we revisit the theory for intrinsic Josephson effects from standpoints of the couplings between stacked junctions and consequent superconducting phase dynamics. After reviewing the previous simulation works based on the theory, we present recent multi-physics simulation studies suggested by authors to systematically reproduce the emission experiments. Some typical simulation results are exhibited and the future perspective of the simulation is also given.

To model the IV-characteristics of shunted intrinsic Josephson junctions we calculated phase dynamics of a chain of two identical junctions covered by a resistive shunt and with a resistive crosspiece between the shunt and the junctions. The coherent solution of dynamic equations for this system is unstable at some bias currents above critical currents of junctions. We have shown that this instability leads to the multibranch structure of IV-curves in the hysteretic region. The multibranch structure appears due to the random switching of junctions to the superconducting state above their critical currents. We found that the resistance of the interface between a covering shunt and the junctions can regulate the multibranch behaviour.

The physics of pizza tossing and the next generation of micromotors Is diamagnetism possible classically? Phase dynamics of two coupled oscillators under external periodic force Optical communications through dense matter

To reveal the dynamic mechanism underlying Parkinsonian resting tremor, we applied a phase dynamics modelling technique to local field potentials and accelerometer signals recorded in three Parkinsonian patients with implanted depth electrodes. We detect a bidirectional coupling between the subcortical oscillation and the tremor. The tremor → brain driving is a linear effect with a small delay corresponding to the neural transmission time. In contrast, the brain → tremor driving is a nonlinear effect with a long delay in the order of 1–2 mean tremor periods. Our results are well reproduced for an ensemble of 41 tremor epochs in three Parkinsonian patients and confirmed by surrogate data tests and model simulations. The uncovered mechanism of tremor generation suggests to specifically counteract tremor by desynchronizing the subcortical oscillatory neural activity.

This paper discusses the excitations in a condensed Bose system as manifestations of the dynamics of the phase. It is argued that the standard approach based on spontaneous symmetry breaking is not viable; a superselection rule enforces strict particle conservation. Some recent theoretical results are used to show how the significance of the phase dynamics can be retained without violating the superselection rule.

Kinetics of phase changes often involves states with broken translational symmetry typified by domain walls which may generally be called patterns. Here we present a brief review of our works in this area emphasizing pattern evolution in incommensurate modulated systems. We show that the idea of phase dynamics introduced recently in hydrodynamics combined with defect dynamics is very useful in describing slightly disturbed modulated states. Otherwise, the equations of motion of discommensurations and dislocations must be solved to obtain pattern evolution.