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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.

Mycobacterium tuberculosis can exist within a host for lengthy periods, tolerating even antibiotic challenge. This non-heritable, antibiotic tolerant "persister" state, is thought to underlie latent Tuberculosis (TB) infection and a deeper understanding thereof could inform treatment strategies. In addition to experimental studies, mathematical and computational modelling approaches are widely employed to study persistence from both an in vivo and in vitro perspective. However, specialized models (partial differential equations, agent-based, multiscale, etc.) rely on several difficult to determine parameters. In this study, a dynamic mathematical model was developed to predict the response of Mycobacterium smegmatis (a model organism for M. tuberculosis) grown in batch culture and subjected to a range of in vitro environmental stresses. Lag phase dynamics, pH variations and internal nitrogen storage were mechanistically modelled. Experimental results were used to train model parameters using global optimization, with extensive subsequent model validation to ensure extensibility to more complex modelling frameworks. This included an identifiability analysis which indicated that seven of the thirteen model parameters were uniquely identifiable. Non-identifiable parameters were critically evaluated. Model predictions compared to validation data (based on experimental results not used during training) were accurate with less than 16% maximum absolute percentage error, indicating that the model is accurate even when extrapolating to new experimental conditions. The bulk growth model can be extended to spatially heterogeneous simulations such as an agent-based model to simulate in vitro granuloma models or, eventually, in vivo conditions, where distributed environmental conditions are difficult to measure.

Under consideration is the system of partial differential equations describing the dynamics of a two-phase medium. Exact partially invariant solutions of rank 1 and defect 1 of this system are obtained with respect to some four-dimensional subalgebras. The phenomenon of collapse (an instantaneous source) in a two-phase medium is described.

The problem of interval estimation of the time delay of the coupling between oscillatory systems from observed time series is considered. It is shown that the known asymptotic estimates based on the empirical model in the form of a system of first-order phase oscillators and the maximum likelihood formalism can lead to false inferences of the value of the time delay in two typical situations: 1) nonlinear low-dimensional systems whose phases are well-determined but, as a result of significant amplitude fluctuations, the phase approximation is insufficient for describing the dynamics and 2) systems whose phases are defined not quite well because of too large amplitude fluctuations. A method for empirical diagnostics of problematic situations and its modification (coarse estimation) providing a low probability of false inferences in these situation are proposed. The efficiency of the diagnostic criterion and coarse estimation suggested is demonstrated on reference systems with different dynamic properties (linear stochastic oscillators, van der Pol oscillators, and Ressler and Lorenz chaotic systems).

While Boolean logic has been the backbone of digital information processing, there exist classes of computationally hard problems wherein this paradigm is fundamentally inefficient. Vertex coloring of graphs, belonging to the class of combinatorial optimization, represents one such problem. It is well studied for its applications in data sciences, life sciences, social sciences and technology, and hence, motivates alternate, more efficient non-Boolean pathways towards its solution. Here we demonstrate a coupled relaxation oscillator based dynamical system that exploits insulator-metal transition in Vanadium Dioxide (VO

We examine the impact of small parity-time () symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing perturbations offer potentially novel means for the control of light in honeycomb lattices.

In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter α, states of traveling phase-domain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them.

Characterizing the complicated flow behaviors arising from high water cut and low velocity oil-water flows is an important problem of significant challenge. We design a high-speed cycle motivation conductance sensor and carry out experiments for measuring the local flow information from different oil-in-water flow patterns. We first use multivariate time-frequency analysis to probe the typical features of three flow patterns from the perspective of energy and frequency. Then we infer complex networks from multi-channel measurements in terms of phase lag index, aiming to uncovering the phase dynamics governing the transition and evolution of different oil-in-water flow patterns. In particular, we employ spectral radius and weighted clustering coefficient entropy to characterize the derived unweighted and weighted networks and the results indicate that our approach yields quantitative insights into the phase dynamics underlying the high water cut and low velocity oil-water flows.

The need to estimate the delay times between elements of coupled oscillatory ensembles is frequently encountered in the analysis of systems of various natures, for which only separate records (time series) of the observed oscillations are available. The possibility of obtaining these estimates in the case of relatively short time series recorded in the presence of noise is considered. Using the well-known point estimator based on a model of the phase dynamics, an interval estimator is proposed, a correction that eliminates a bias of the point estimate is introduced, and a formula for the confidence interval is derived based on the maximum likelihood formalism. Possibilities of the proposed approach are illustrated by the results of numerical experiments in systems of coupled oscillators.

We develop a systematic method for deriving the phase dynamics of perturbed periodic solutions. The method is to regard periodic solutions as slowly modulated traveling solutions on the circle. There, problems are reduced to the perturbed problems from stationary solutions on the circle. This makes the treatment of periodic solutions far easier and systematic. We also give the rigorous proofs for this method.

Based on the model system undergoing phase separation and chemical reactions, we investigate the dynamics of propagating dissipative waves under external forcing which is periodic both in space and time. A phase diagram for the entrained and non-entrained states under the external forcing is obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carried out to show that the transition between the entrained and the non-entrained states by changing the external frequency occurs either through a saddle-node bifurcation or through a Hopf bifurcation and that these two bifurcation lines are connected at a Bogdanov-Takens bifurcation point.

We explore identical R\"ossler systems organized into two equally-sized groups, among which differing positive and negative in- and out-coupling strengths are allowed. Patterns of distinctly synchronized phase dynamics are observed, which coexist with chaotically evolving amplitudes. In particular, we report the emergence of traveling phase waves, i.e. states in which the oscillators settle on a new rhythm different from their own. We further elucidate our findings through phase-coupled R\"ossler systems, establishing a connection with the Kuramoto model. Together with the study of noise effects, our results suggest a promising new avenue towards the coexistence of chaotic, noisy and regular collective dynamics., Comment: 5 pages, 4 figures and Supplemental Material

We study a learning and recalling model of phase patterns in a two- or three-coupled BVP oscillators system with a time delay δ. The coupling strengths are modulated by the Hebbian learning rule. Assuming the first-order approximation, we calculate the optimal condition of δ for exact recall by applying the phase dynamics theory. When α = 0, where α represents the coupling from activator to inhibitor, the correlation between the learning and the retrieval phase depends on δ. When α = 1, exact recall is achieved independent of δ. The results can be explained by the phase dynamics theory.

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

The statistics of the instantaneous phase of oscillations in dynamic systems with a noisy limit cycle is compared to the statistics of the instantaneous phase of oscillations in dynamic systems with a spiral chaotic attractor. Simulation of the phase dynamics of chaotic self-sustained oscillations by a Wiener process is considered. The results provided by various methods of determination of the instantaneous phase are analyzed.

A vector equation of motion in the Lagrangian frame of reference is rearranged into a set of two scalar equations to determine the relative-velocity magnitude of a particle and its direction. The resulting system, written in terms of the Lame coefficients, is solved jointly with continuum equations of motion in the Eulerian frame of reference. The method is exemplified by the flow of a heterogeneous mixture between permeable parallel walls and the flow of a heterogeneous non-Newtonian fluid over the surface of a sedimentation centrifuge rotor.

For αβ>−1, stable time periodic solutions A(X,T)=AqeiqX+iωqT are the locally preferred planform for the complex Ginzburg–Landau equation ∂ T A=A+(1+iα)∂ X 2 A−(1+iβ)A|A| 2 . In order to describe the spatial global behavior, an evolution equation for the local wave number q can be derived formally. The local wave number q satisfies approximately a conservation law ∂τq=∂ξh(q). It is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation.

In this paper we extendthe results of Frankel andKiemel ( SIAM J. Appl. Math ,5 3 (1993), pp. 1436-1446) to a network of slowly coupledoscillators. First, we use Malkin's theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupledoscillators (1) near And ronov-Hopf bifurcations, (2) near saddle-node on invariant circle bifurcations, and (3) near relaxation oscillations. We compare andcontrast synchronization properties of slowly andweakly coupledoscillators.

Soap froths as archetypal disordered cellular structures, exhibiting spatial and temporal evolution, have been predominantly studied in terms of their topological properties. Recently, constrained geometries in froth have attracted attention, since these arise both from existing or natural structure or, more usually, can be artificially configured. Using a direct simulation method, which retains detailed information on bubble (cell) mechanisms, we have investigated dynamics in 2D froths with various initial structures corresponding to controlled disorder. In particular, we examine the special case of a defect ring surrounding a central inclusion in a uniform froth, for different number of defects and ring radius. This geometry permits comparison with shell theory, as well as insight on dynamics of a virtual phase (as defined for persistence and non-equilibrium processes in general). It appears that defect location and pattern of inclusion in the virtual phase cause considerable variation in the evolutionary behaviour, leading to non-universal exponents for the phase dynamics. This is probably explained by the fact that the froth is still in the transient period over simulation time scales, rather than achieving the final stage of persistence. However, distinctive patterns of response can be identified for the different froth regions, despite system size limitations, with topological and other properties indicating that a quasi-equilibrium results.

Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in \emph{Phys. Rev. Lett.} \textbf{111}, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincar\'e oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems., Comment: 9 pages, 8 figures

The objective of the present paper is the modeling and analysis of the dynamics of macrophages and certain growth factors in the inflammatory phase, the first one of the wound healing process. It is the phase where there exists a major difference between diabetic and non-diabetic wound healing, an effect that we will consider in this paper Supported by grants MTM2011-27739-C04-01 (Spain) and 2009SGR345 (Catalonia). Supported by FAPESP 2008/55516-3 and CAPES/DGU 238/11 (Brazil). Supported by grants MTM2011-27739-C04-03 (Spain) and 2009SGR345 (Catalonia)

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 investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data. A description of the regularizing phenomena is given in terms of the corresponding space-time scale.

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.

We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations., This paper has been withdrawn by the author due to errors in Fig. 3 and faulty conclusions drawn from Fig. 4

Cross-spectral and synchronization analysis of two independent, identical chaotic R\"ossler systems suggest a coupling although there is no interaction. This spuriously detected interaction can either be explained by the absence of mixing or by finite size effects. To decide which alternative holds the phase dynamics is studied by a model of the fluctuations derived from the system's equations. The basic assumption of the model is a diffusive character for the system which corresponds to mixing. Comparison of theoretical properties of the model with empirical properties of the R\"ossler system suggests that the system is mixing but the rate of mixing appears to be rather low.

Coding for the additive white Gaussian noise (AWGN) channel that also introduces an unknown carrier phase rotation is considered in this work. A simple model that captures the phase dynamics is one, where the unknown carrier phase is considered constant over a block of N symbols, and independent from block to block. In this work, LDPC codes in conjunction with pilot symbols are considered. In particular, a single pilot symbol of specified power is inserted in the beginning of each block of length N, in order to aid the joint estimation and decoding process. As expected, the presence of a pilot symbol implies an inevitable loss in the transmitted power, resulting in a trade-off between the power allocated between the pilots and the coded bits, and the quality of the joint estimation and decoding process.

The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and the Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and the zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wavenumbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings.