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Spontaneous contractions of cardiomyocytes are driven by calcium oscillations due to the activity of ionic calcium channels and pumps. The beating phase is related to the time-dependent deviation of the oscillations from their average frequency, due to noise and the resulting cellular response. Here, we demonstrate experimentally that, in addition to the short-time (1--2 Hz), beat-to-beat variability, there are long-time correlations (tens of minutes) in the beating phase dynamics of isolated cardiomyocytes. Our theoretical model relates these long-time correlations to cellular regulation that restores the frequency to its average, homeostatic value in response to stochastic perturbations.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit., 16 pages, 8 figures

This paper analyzes the injection-locking and injection-pulling behavior of two mutually coupled single-loop free-running OEOs with nearby frequencies. We derive the phase dynamics equations of two mutually-coupled OEOs considering weak RF signal injection and the phase stability of the mutually injection-locked system is studied. Also, we present compact closed form expressions for the oscillation frequencies of two mutually injection-pulled OEOs in terms of the injection-ratios and the phase shifts of the mutually injected RF signals. The beat frequency of the mutually injection-pulled system is calculated in terms of the injection ratios. The influence of mutual injection-pulling on the RF output power spectrums of two OEOs are studied. It is shown that the RF output power spectrums of two mutually injection-pulled OEOs are highly asymmetric and strongly depends on the injection ratios. The experimental results corroborating the theoretical findings are given.

First order phase reduced model is a good approximation of the dynamics of forced nonlinear oscillators near its limit cycle. The phase evolution is determined by the unforced frequency, the forcing term, and the phase response curve (PRC). Such models arise in biological oscillations such as in circadian rhythm, neural signaling, heart beat, etc. This technical note focuses on the phase regulation of the circadian rhythm using light intensity as the input. Though the model is simple, the circle topology of the state space needs to be carefully addressed. The most common entrainment method is to use a periodic input, such as in our daily light-dark cycle. We obtain the complete stable entrainment condition based on the entraiment input and the PRC. Motivated by the jet-lag problem, we also consider the minimum time entrainment control to achieve a specified phase shift. Application of the Pontryagin Minimum Principle leads to an efficient solution strategy for the optimal control, without solving the two-point boundary value problem. The optimal control may be further represented as a feedback control law based on the current and desired phases. Our analysis allows the answer to questions such as: When traveling from New York to Paris, is it faster to use light to shift the phase forward by 6 hours or delay the phase by 18 hours? The answer is somewhat counter-intuitive—delaying by 18 hours requires less time. The general answer depends on the light intensity level and the shape of the PRC. PRCs for human and Drosophila from the literature are used to illustrate the results.

Phase identification is an arduous task during high-throughput processing experiments, which can be exacerbated by the need to reconcile results from multiple measurement techniques to form a holistic understanding of phase dynamics. Here, we demonstrate AutoPhase, a machine learning algorithm, which can identify the presence of the different phases in spectral and diffraction data. The algorithm uses training data to determine the characteristic features of each phase present and then uses these features to evaluate new spectral and diffraction data. AutoPhase was used to identify oxide phase growth during a high-throughput oxidation study of NiAl bond coats that used x-ray diffraction, Raman, and fluorescence spectroscopic techniques. The algorithm had a minimum overall accuracy of 88.9% for unprocessed data and 98.4% for postprocessed data. Although the features selected by AutoPhase for phase attribution were distinct from those of topical experts, these results show that AutoPhase can substantially increase the throughput high-throughput data analysis.

Although the phase shifts in ongoing oscillations seen in electroencephalograms (EEGs) and magnetoencephalograms are an important factor in discussions of phase dynamics, such as synchrony and reset, few studies have focused specifically on the phase shift. Here we investigate the relationship between phase shifts in alpha-frequency rhythms and reaction times during a visual simple reaction task by applying our previously described method (Naruse et al., 2013), which enables detection of phase shifts from a single EEG trial. In the left, parietal, and occipital areas, the reaction times in the trials in which phase shifts were detected before the button press were significantly longer than in those in which phase shifts were not so detected. These results indicate that phase shifts in the alpha and mu rhythms relate to variability in reaction times.

A dynamic system showing stable rhythmic activity can be represented by the dynamics of phase oscillators. This would provide a useful mathematical framework through which one can understand the system's dynamic properties. A recent study proposed a Bayesian approach capable of extracting the underlying phase dynamics directly from time-series data of a system showing rhythmic activity. Here we extended this method to spike data that otherwise provide only limited phase information. To determine how this method performs with spike data, we applied it to simulated spike data generated by a realistic neuronal network model. We then compared the estimated dynamics obtained based on the spike data with the dynamics theoretically derived from the model. The method successfully extracted the modeled phase dynamics, particularly the interaction function, when the amount of available data was sufficiently large. Furthermore, the method was able to infer synaptic connections based on the estimated interaction function. Thus, the method was found to be applicable to spike data and practical for understanding the dynamic properties of rhythmic neural systems.

We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct numerical simulations of the three models., Comment: 19 pages, 11 figures

We consider networks of coupled phase oscillators of different complexity: Kuramoto–Daido-type networks, generalized Winfree networks, and hypernetworks with triple interactions. For these setups an inverse problem of reconstruction of the network connections and of the coupling function from the observations of the phase dynamics is addressed. We show how a reconstruction based on the minimization of the squared error can be implemented in all these cases. Examples include random networks with full disorder both in the connections and in the coupling functions, as well as networks where the coupling functions are taken from experimental data of electrochemical oscillators. The method can be directly applied to asynchronous dynamics of units, while in the case of synchrony, additional phase resettings are necessary for reconstruction.

Phase plays a crucial role in many quantum effects including interference. Here we lay the foundations for the study of phase in probabilistic theories more generally. Phase is normally defined in terms of complex numbers that appear when representing quantum states as complex vectors. Here we give an operational definition whereby phase is instead defined in terms of measurement statistics. Our definition is phrased in terms of the operational framework known as generalized probabilistic theories or the convex framework. The definition makes it possible to ask whether other theories in this framework can also have phase. We apply our definition to investigate phase and interference in several example theories: classical probability theory, a version of Spekkens' toy model, quantum theory and box-world. We find that phase is ubiquitous; any non-classical theory can be said to have non-trivial phase dynamics. © IOP Publishing and Deutsche Physikalische Gesellschaft.

A piezoelectric tube is shown to have linear, but non-minimum phase dynamics. The main impediment to the actuation of this piezoelectric tube is the presence of a low-frequency resonant mode which causes mechanical vibrations. A physical-model-based control method is extended to non-minimum phase systems in general and successfully applied to damp the resonant mode; leading to a vibration-free actuation of the piezoelectric tube.

Multi-electrode recordings of local field potentials (LFPs) provide the opportunity to investigate the spatiotemporal organization of neural activity on the scale of several millimeters. In particular, the phases of oscillatory LFPs allow studying the coordination of neural oscillations in time and space and to tie it to cognitive processing. Given the computational roles of LFP phases, it is important to know how they relate to the phases of the underlying current source densities (CSDs) that generate them. Although CSDs and LFPs are distinct physical quantities, they are often (implicitly) identified when interpreting experimental observations. That this identification is problematic is clear from the fact that LFP phases change when switching to different electrode montages, while the underlying CSD phases remain unchanged. In this study we use a volume-conductor model to characterize discrepancies between LFP and CSD phase-patterns, to identify the contributing factors, and to assess the effect of different electrode montages. Although we focus on cortical LFPs recorded with two-dimensional (Utah) arrays, our findings are also relevant for other electrode configurations. We found that the main factors that determine the discrepancy between CSD and LFP phase-patterns are the frequency of the neural oscillations and the extent to which the laminar CSD profile is balanced. Furthermore, the presence of laminar phase-differences in cortical oscillations, as commonly observed in experiments, precludes identifying LFP phases with those of the CSD oscillations at a given cortical depth. This observation potentially complicates the interpretation of spike-LFP coherence and spike-triggered LFP averages. With respect to reference strategies, we found that the average-reference montage leads to larger discrepancies between LFP and CSD phases as compared with the referential montage, while the Laplacian montage reduces these discrepancies. We therefore advice to conduct analysis of two-dimensional LFP recordings using the Laplacian montage. RH and GD were funded by the European Research Council (Advanced Grant DYSTRUCTURE No. 295129), the Spanish Research Project PSI2013-42091-P, the CONSOLIDER-INGENIO 2010 Program CSD2007-00012, and the FP7-ICT Brainscales (269921). XA and PV are supported by the European Research Council's CDAC project: “The Role of Consciousness in Adaptive Behavior: A Combined Empirical, Computational and Robot based Approach” (ERC-2013- ADG 341196).

In this paper, we investigate various aspects of electro-magnetically induced transparency (EIT) that are associated with quantum interference. In the first half of this paper, we investigate two cascade schemes and demonstrate two possible absorption pathways in one, which leads to interference, and only one pathway in the other scheme, which does not exhibit EIT. In the second part of this paper, we demonstrate how EIT can be changed into enhanced absorption by changing the phase of either the coupling or probe fields.

We study control of synchronization in weakly coupled oscillator networks by using a phase reduction approach. Starting from a general class of limit cycle oscillators we derive a phase model, which shows that delayed feedback control changes effective coupling strengths and effective frequencies. We derive the analytical condition for critical control gain, where the phase dynamics of the oscillator becomes extremely sensitive to any perturbations. As a result the network can attain phase synchronization even if the natural interoscillatory couplings are small. In addition, we demonstrate that delayed feedback control can disrupt the coherent phase dynamic in synchronized networks. The validity of our results is illustrated on networks of diffusively coupled Stuart--Landau and FitzHugh--Nagumo models.

A new method is proposed for the investigation of transitions from synchronous dynamics of coupled oscillators to asynchronous regimes attained upon synchronization breakage by a sequence of short phase resetting pulses. The possibility to use the proposed method for determining the coefficient of coupling between interacting systems from an analysis of the signals generated by these systems is considered.

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.

We construct a theory of dynamical behavior in intrinsic Josephson junction stacks with multigap superconducting layers. The theory predicts the existence of two kinds of phase modes, one of which is the Josephson-plasma mode and other of which is the Leggett’s mode. We discuss a cooperative phenomena induced by inter-band Josephson coupling in addition to capacitive and inductive couplings between the superconducting layers.

Based on neurophysiological evidence, studies modeling human locomotion system have shown that a bipedal walking is generated by mutual entrainments between the oscillatory activities of a central pattern generator (CPG) and a Body. The walking model could well reproduce human walking. However, it has been also shown that time delay in the sensorimotor loop destabilizes mutual entrainments, which leads a failure to walk. Recently, theoretical studies have discovered a phenomenon in which a CPG can induce the phase of its oscillatory activity to shift forward according to time delay. This self-organized phenomenon overcoming time delay is called “flexible-phase locking”. Then, theoretical studies have hypothesized that one of the essential mechanisms to yield of flexible-phase locking is a stable limit cycle of CPG activity. This study demonstrates the hypothesis in walking models through computer simulation by replacing the CPG model with the one having different oscillation properties.

We present a novel approach for recovery of the directional connectivity of a small oscillator network by means of the phase dynamics reconstruction from multivariate time series data. The main idea is to use a triplet analysis instead of the traditional pairwise one. Our technique reveals an effective phase connectivity which is generally not equivalent to a structural one. We demonstrate that by comparing the coupling functions from all possible triplets of oscillators, we are able to achieve in the reconstruction a good separation between existing and non-existing connections, and thus reliably reproduce the network structure.

We consider a $\ensuremath{\varphi}$ Josephson junction, which has a bistable zero-voltage state with the stationary phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. In the nonzero voltage state the phase ``moves'' viscously along a tilted periodic double-well potential. When the tilting is reduced quasistatically, the phase is retrapped in one of the potential wells. We study the viscous phase dynamics to determine in which well ($\ensuremath{-}\ensuremath{\varphi}$ or $+\ensuremath{\varphi}$) the phase is retrapped for a given damping, when the junction returns from the finite-voltage state back to the zero-voltage state. In the limit of low damping, the $\ensuremath{\varphi}$ Josephson junction exhibits a butterfly effect---extreme sensitivity of the destination well on damping. This leads to an impossibility to predict the destination well.

The moderately damped regime (MDR) in a Josephson junction (JJ) is quite distinct from the well-known case of underdamped systems and quite common in devices characterized by low critical currents. Measurements of switching current distribution (SCD) are a direct way of discriminating the phase dynamics and the transport also in nontrivial cases of moderate damping. MDR is going to be more and more common with advances in nanopatterning superconductors and in materials science finalized to build hybrid devices. We report on measurements of SCDs, both in the thermal and quantum regime, on moderately damped NbN/MgO/NbN JJs. A transition from the Thermal Activation (TA) regime to Macroscopic Quantum Tunnelling (MQT) occurs at about 90 mK, while above 1.6 K a phase diffusive dynamics is clearly observed. The combined analysis of these regimes through numerical simulations allowed us to extract the fundamental parameters, which completely characterize the phase dynamics of such moderately damped JJs.

International audience; Motivated by neuroscience applications, we rigorously derive the phase dynamics of an ensemble of interconnected nonlinear oscillators under the effect of a proportional feedback. We individuate the critical parameters determining the validity of the phase approximation and derive bounds on the accuracy of the latter in reproducing the behavior of the original system. We use these results to study the existence of oscillating phase-locked solutions in the original oscillator model.

In this paper, we propose a framework for generating coordinated periodic movements of robotic systems with external inputs. We developed an adaptive pattern generator model that is composed of a two-factor observation model with a style parameter and phase dynamics with a phase variable. The style parameter controls the spatial patterns of the generated trajectories, and the phase variable controls its temporal profiles. To validate the effectiveness of our proposed method, we applied it to a simulated humanoid model to perform biped walking behaviors coordinated with observed walking patterns and the environment. The robot successfully performed stable biped walking behaviors even when the style of the observed walking pattern and the period were suddenly changed.

Previous studies of the phase properties of optical linear amplifiers and attenuators have been restricted to situations where the effect on the phase of single-mode light is found to be diffusive. In this paper we give an analysis of phase-sensitive linear amplification and attenuation in the weak-field regime that clearly reveals a non-diffusive effect on the phase. Indeed we show that a weak field which has initially a uniform phase probability density can inherit non-random phase properties from both phase-sensitive attenuators and amplifiers. In contrast, we find that phase-insensitive amplifiers and attenuators have merely a diffusive effect on the phase of the light.

The phase jump dynamics in a photosensitive Belousov-Zhabotinsky reaction has been studied. It was found that an initially inhomogeneous phase distribution evolved to a homogeneous one, so that the phase at phase-lag points rises to the value at phase-lead points. A steep phase distribution was shown to become smoother in the course of time, inducing a dephasing wave which restores the system to a uniform phase. Applications of photosensitive chemical reactions to image processing (image storage, inversing, contouring) are discussed

Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time.

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.

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.

In this paper we present an electronic system designed to emulate neural networks. Two major restrictions are assumed: discrete synapses (+1,0,−1) and threshold-type neurons. Drawbacks given by restrictions are solved with a more complex learning algorithm that maps real valued configurations for synapses into bipolar ones. The central unit of the system is an ASIC designed in accordance with a new sequential dynamics, which is faster and with better recall characteristics. As a result, the system designed has a fast recall phase and a large number of neurons. A handwritten OCR system is being designed.

We study the relationship between synchronization and the rate with which information is exchanged between nodes in a spatio-temporal network that describes the dynamics of classical particles under a substrate Remoissenet-Peyrard potential. We also show how phase and complete synchronization can be detected in this network. The difficulty in detecting phase synchronization in such a network appears due to the highly non-coherent character of the particle dynamics which unables a proper definition of the phase dynamics. The difficulty in detecting complete synchronization appears due to the spatio character of the potential which results in an asymptotic state highly dependent on the initial state., to appear in PRAMANA

We have studied phase dynamics in EIT. We observed how the transmission decreases and recovers when the phase of the optical field changes abruptly. The result can be used for the fast control of EIT.

Based on the complex Ginzburg-Landau equation (CGLE), a new mapping model of oscillatory media is proposed. The present dynamics is fully determined by an effective phase field renormalized by amplitude. The model exhibits phase turbulence, amplitude turbulence, and a frozen state reported in the CGLE. In addition, we find a state in which the phase and amplitude have spiral structures with opposite rotational directions. This state is found to be observed also in the CGLE. Thus, one concludes that the behaviors observed in the CGLE can be described by only the phase dynamics appropriately constructed.

A phased-array two-dimensional CPLL array is introduced. The phase dynamics are analyzed and an approximate solution are obtained using the method of continuum phase. By detuning the free-running frequencies of edge elements, linear phase distribution can be achieved among the array. An experimental CPLL array circuit is also shown and tested, proceedings.

We study the model of a massive inflaton field $\phi$ coupled to another scalar filed $\chi$ with interaction term $g^2\phi^2\chi^2$ for the first stage of preheating. We obtain the the behavior of the phase in terms of the iteration of a simple family of circle maps. When expansion of the universe is taken into account the qualitative behavior of the phase and growth number evolution is reminiscent of the behavior found in the case without expansion., Comment: 4 pages, 4 figures, LaTeX; submitted to the Proceedings of Eleventh Marcel Grossmann Meeting

Synchronization phenomena in coupled circadian oscillators of plant leaves were investigated experimentally using bioluminescence technology for a clock gene. Analyzing the phase of circadian oscillation, the phase-wave propagations and the phase delay caused by the vein network were observed. We describe these phase dynamics using a two-layer model with coupled Stuart-Landau equations. Global synchronization of circadian oscillators in the leaf is also investigated.

We present results from our time dependent model on electro-magnetically induced transparency in a standard Λ configuration. We show that sudden phase jumps in either field leads to transient enhanced absorption for the probe field.

Coherent terahertz (THz) radiations based on intrinsic Josephson junctions (IJJs) from a Bi2Sr2CaCu2O[Formula: see text] (BSCCO) mesa under a bias voltage were observed in 2007. Later on, a novel [Formula: see text] phase kink state was proposed which can explain the key experimental results. However, how to drive the system into the [Formula: see text] phase kink state efficiently still remains an open problem. In this work, we propose several ways to synchronize the system to the [Formula: see text] phase kink state. We start from random and typical regular initial phase configurations. It is found that starting from some initial phase configurations, the system can be driven efficiently into the [Formula: see text] phase kink state. The time spent before entering the [Formula: see text] phase kink states for these initial phase configurations is shorter when the initial phase configuration is further away from the uniform phase configuration.

This paper describes a fully distributed communication timing control method for wireless sensor networks, which exploits the properties of the emergent behavior of coupled phase oscillatory dynamics. Assuming the periodic data communications in massively distributed and locally interacting sensor nodes, a phase oscillatory dynamics is employed to realize a collision free communication timing control. For this purpose, we propose Phase Diffusion Synch- Alliance (PDSA) method, where coordination in repulsive and synchronous actions in the phase dynamics will take an essential role to self-organize a phase difference pattern corresponding to an efficient communication timing pattern in the distributed sensor nodes. We formulate the method and conduct simulations to verify effectiveness of PDSA.

We measure the phase dynamics and statistics of SOA-based wavelength conversion and XOR operation using linear optical sampling. We elucidate the role of the pattern dependent phase fluctuation in limiting the performance of XOR operation. (2 pages)

The relative phase between two uncoupled Bose-Einstein condensates tends to attain a specific value when the phase is measured. This can be done by observing their decay products in interference. We discuss exactly solvable models for this process in cases where competing observation channels drive the phases to different sets of values. We treat the case of two modes which both emit into the input ports of two beam splitters and of a linear or circular chain of modes. In these latter cases, the transitivity of the relative phase becomes an issue.

Focusing on the mechanisms underlying the appearance of phase multistability, the paper examines a variety of phase-locked patterns in the bursting behavior of a model of coupled pancreatic cells. In particular, we show how the number of spikes per train and the proximity of a neighboring equilibrium point can influence the formation of coexisting regimes.

We study numerically the effects of noise and periodic forcings on cluster synchronization in a chain of Van der Pol oscillators. We generalize the notion of effective synchronization to the case of a spatially extended system. It is shown that the structure of synchronized clusters can be effectively controlled by applying local external forcings. The effect of amplitude relations on the phase dynamics is also explored.

In this note we will introduce in the first section the concept of phase dynamics, followed by an outline of some recent developments associated with the properties of nonlinear phase equations, and we will conclude with an extensive list of references on phase dynamics and closely related subjects including envelope equations, the motion of defects, and the coupling between amplitude and phase equations.

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.

Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both. We would like to thank Mark Woolrich for his contribution to data acquisition and analysis and the fruitful discussion about interpretation of our findings. We received the following funding: GD, RT: ERC Advanced Grant: DYSTRUCTURE (n. 295129), GD: Spanish Research Project PSI2013-42091-P, Human Brain Project, and FP7-ICT BrainScales (269921). MLK: ERC Consolidator Grant CAREGIVING (n. 615539) and Center for Music in the Brain/Danish National Research Foundation (DNRF117). AD, BP: H2020-MSA-ITN COSMOS (n. 642563).

We present the nonlinear phase equations describing the stability of a one-dimensional periodic pattern, with mean flow effect due to Galilean invariance. We show that the phase dynamics is second order in time, and could lead to an oscillatory instability of the pattern.

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.