In this paper we present a comprehensive analysis of line tension-driven compositional interface fluctuations in planar lipid bilayer membranes. Our starting point is the advective Cahn-Hilliard equation for the local lipid composition in symmetric membranes, which explicitly incorporates both advective and diffusive lipid transport processes, and which is coupled to the continuum hydrodynamic equations governing the flow behavior of the membrane and surrounding solvent with finite subphase thickness. In order to extract the interface dynamics from the continuum phase-field formalism, we first derive the appropriate sharp-interface limit equations. We then carry out a linear perturbation analysis for the relaxational dynamics of small-amplitude sinusoidal interface fluctuations to yield the general dispersion relation ${\omega }_k$ as a function of perturbation wave number $k$. The resulting expression incorporates the effects of diffusive and advective lipid transport processes within the membrane, viscous or viscoelastic membrane properties, coupling between membrane and solvent, and inertial effects within the membrane and solvent. It is shown that previously considered scenarios naturally emerge as limiting cases of the general result. Furthermore, we discuss two additional scenarios amenable to analysis, one in which the inertia of the solvent is relevant, and another one in which the membrane displays significant viscoelastic properties. Finally, we numerically evaluate the general dispersion relation for three representative model membrane systems.

Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well as chaotic oscillations. We have experimentally observed these synchronized states in delay-coupled electronic circuits and have analyzed their stability by numerical simulations and analytical calculations. We found that the conditions for longitudinal and transversal stability largely exclude each other and prevent, e.g., the synchronization of Pyragas-controlled orbits. Most striking is the observation of complete chaotic synchronization for large delay times, which should not be allowed in the given coupling scheme on the background of the actual paradigm.

Convective instabilities in a cylindrical cavity heated from below, with a free surface at the top, are numerically investigated using a spectral-element code. Both buoyancy and surface tension forces are taken into account, and heat exchange is considered at the upper surface. This configuration corresponds to the Bénard-Marangoni situation. The primary thresholds associated with azimuthal eigenmodes and corresponding to the onset of convection are first given as a function of the aspect ratio of the cavity $A$ (radius/height), the Biot number $\mathrm{Bi}$, and the Marangoni number $\mathrm{Ma}$. Particular attention is paid to the influence of the Biot and Marangoni numbers: a stabilizing surface tension effect ($\mathrm{Ma}>0$) induces an increase of the primary thresholds, which is magnified for small values of $\mathrm{Bi}$, but may also change the flow structure by creating counter-rotating rolls near the free surface. The nonlinear evolution of the convection beyond its onset is given through bifurcation diagrams for $A=1.5$. Two different branches of axisymmetric solutions, either with upflow or downflow at the center, emerge at the onset. The destabilization of these solutions and the further dynamical evolution of the flow has been highlighted for widely varying Biot numbers.

The effect of a constant and uniform horizontal magnetic field on the flow in a cylindrical cavity heated from below, with a free surface at the top, is numerically investigated. The azimuthal modes, which usually trigger convection in a cylinder, are changed by the horizontal magnetic field to oriented modes, either parallel or perpendicular to the magnetic field direction. The corresponding primary thresholds increase with the Hartmann number $\mathrm{Ha}$. This increase, however, depends on the structure of the modes and is the weakest for the parallel modes and the strongest for the perpendicular modes. The changes that affect the evolution of the primary thresholds with the aspect ratio for nonzero $\mathrm{Ha}$ are also emphasized. The nonlinear evolution of the convection with a horizontal magnetic field is presented through bifurcation diagrams for different values of the Prandtl number $\mathrm{Pr}$. For $\mathrm{Pr}=1$ and small values of $\mathrm{Ha}$, the structuring effect of the horizontal magnetic field, which involves modifications of the flow structures and bifurcation points, is put into light. Results are finally shown for smaller $\mathrm{Pr}$ values corresponding to liquid metals.

The exact expression for the collective diffusion coefficient in one dimension, obtained by Payne and Kreuzer [Phys. Rev. B. 75, 115403 (2007)], is compared with Monte Carlo simulation. Different hopping kinetics are analyzed. For initial- and final-state interaction kinetics no anomalies are observed. However, for the so-called interaction kinetics where both initial- and final-state interactions are involved, it is shown that even when the transition rates satisfy the principle of detail balance, additional constraints are necessary to guarantee the diffusion of particles. These restrictions give rise to a phase diagram that determines the regions where the exact solution of the diffusion coefficient seem to be not physically sound. The Monte Carlo simulation allows us to analyze the mechanism of diffusion in these regions, where in some cases the simulation does not match the exact solution. A possible explanation is presented.

Rheological properties of soft materials are often investigated in oscillatory shear and characterized by the storage and loss modulus, $G^{\prime }$ and $G^{\prime \prime }$, respectively. Unfortunately, the relaxation dynamics of most soft materials is too slow to be directly probed by commercial rheometers. Recently, it was shown by Wyss et al. [Phys. Rev. Lett. 98, 238303 (2007)] that the application of an oscillating strain-rate drives such soft materials and shifts the structural relaxation to higher times. They called this experimental technique strain-rate frequency superposition (SRFS). The great benefit of SRFS is the extremely extended frequency range. As viscoelastic measures, Wyss et al. proposed the familiar storage and loss modulus. Using these moduli results in a serious drawback: When the material yields, nonlinearities appear and the physical interpretation of the storage and loss modulus breaks down. Thus, SRFS as proposed by Wyss et al. is limited to the linear regime and the benefit of the extended frequency regime vanishes. In the present work, we validate an alternative data analysis technique, recently established as the stress decomposition method [K. S. Cho et al., J. Rheol. 49, 747 (2005); R. H. Ewoldt et al., J. Rheol. 52, 1427 (2008)], for combination with SRFS. Use of the stress decomposition method provides a physical interpretation of linear and nonlinear SRFS data in terms of strain stiffening and softening as well as shear thickening and thinning.

A class of Laplacian growth models in the channel geometry is studied using the formalism of tripolar Loewner evolutions, in which three points, namely, the channel corners and the point at infinity, are kept fixed. Initially, the problem of fingered growth, where growth takes place only at the tips of slitlike fingers, is revisited and a class of exact solutions of the corresponding Loewner equation is presented for the case of stationary driving functions. A model for interface growth is then formulated in terms of a generalized tripolar Loewner equation and several examples are presented. It is shown that the growing interface evolves into a steadily moving finger and that tip competition arises for nonsymmetric initial configurations with multiple tips.

We studied the easy axis gliding of a polymer-stabilized nematic liquid crystal. The easy axis of the liquid crystal was slowly reoriented in the presence of an electric field, and the gliding process was approximated to the triple exponential functions. The different dynamics is considered to be related to the morphology of the polymers formed on the surface and in the bulk. The initial orientation of the easy axis was recovered by the elastic restoring force of the polymers after removal of the electric field. The gliding angle and reorientation time were decreased with the longer constituent monomers, and this is considered to be due to the increased anchoring and reduced surface viscosity by the increased fraction of the polymers intersticed in a liquid crystal near the surface.

Canalization is a classic concept in developmental biology that is thought to be an important feature of evolving systems. In a Boolean network, it is a form of network robustness in which a subset of the input signals controls the behavior of a node regardless of the remaining input. It has been shown that Boolean networks can become canalized if they evolve through a frustrated competition between nodes. This was demonstrated for large networks in which each node had $K=3$ inputs. Those networks evolve to a critical steady state at the border of two phases of dynamical behavior. Moreover, the evolution of these networks was shown to be associated with the symmetry of the evolutionary dynamics. We extend these results to the more highly connected $K>3$ cases and show that similar canalized critical steady states emerge with the same associated dynamical symmetry, but only if the evolutionary dynamics is biased toward homogeneous Boolean functions.

The complex Swift-Hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and, as such, admits solutions in the form of traveling waves. The properties of these waves are systematically analyzed and the dynamics associated with sources and sinks of such waves investigated numerically. A number of distinct dynamical regimes is identified and analyzed using appropriate phase equations describing the evolution of long-wavelength instabilities of both the homogeneous oscillating state and constant amplitude traveling waves.

The interaction of two spiral waves with independent frequencies in a bilayer oscillatory medium (one spiral in each layer) and with a symmetric coupling $e$ is studied. If the spirals have different frequencies, the faster spiral is unaffected by the slower one, and the slower can show a variety of behaviors, which depend on $e$ and include, in order of increasing $e$, phase drifting, amplitude modulation, amplitude domination, and phase synchronization. This high-frequency dominance, the asymmetric driving-response effect under the condition of a symmetric coupling, is generic and independent of whether the coupled spiral waves are outwardly rotating or inwardly rotating spirals. If the spirals have identical frequencies, they may even show complete synchronization, parallel drift, or circular drift, depending on the relative rotation direction of the two spirals and their initial separation distance. Comparisons with coupled spirals in monolayer media, previous works on coupled spirals in bilayer systems, and coupled phase oscillators are made.

We study the manner in which the effect of an external drive is transmitted through mutually coupled response systems by examining the phase synchrony between the drive and the response. Two different coupling schemes are used. Homogeneous couplings are via the same variables while heterogeneous couplings are through different variables. With the latter scenario, synchronization regimes are truncated with an increasing number of mutually coupled oscillators in contrast to homogeneous coupling schemes. Our results are illustrated for systems of coupled chaotic Rössler oscillators.

Rate processes are simple and analytically tractable models for many dynamical systems that switch stochastically between a discrete set of quasistationary states; however, they may also approximate continuous processes by coarse-grained, symbolic dynamics. In contrast to limit-cycle oscillators that are weakly perturbed by noise, in such systems, stochasticity may be strong, and topologies more complicated than a circle can be considered. Here we apply a second-order time-dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete-state oscillators coupled or driven through weakly time-dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.

We show that a fundamental feature of the three-level:two-level model, used to describe molecular monomode lasers with a saturable absorber, is the existence of isolas of periodic passive Q-switching (PQS) self-pulsations. A common feature of these closed families of periodic solutions is that they contain regions of stability of the PQS self-pulsation bordered by period-doubling and fold bifurcations, when the control parameter is either the incoherent external pump or the cavity frequency detuning. These findings unveil the fundamental solution structure that is at the origin of the phenomenon known as “period-adding cascades” in our system. Using numerical continuation techniques we determine these isolas systematically, as well as the changes they undergo as secondary parameters are varied.

We study analytically and numerically the linear stability of weakly nonlocal solitons in the parametrically driven nonlinear Schrödinger equation. Two exact solutions are derived in an implicit form. We show analytically that despite the well-known stabilizing properties of nonlocality one of the solitons remains unstable even in the nonlocal case for any values of the dissipation, the damping, and the degree of nonlocality. The second soliton, as compared to its local counterpart, attains wider stable regions in the space of the parameters of the system.

When a polymer solution is dried in air, a polymer-concentrated region, called a “skin” layer, often appears near the surface. In this paper, an experimental method is proposed for detecting the initial process of the formation of the skin layer. An electric field is applied on the surface of polymer solutions by a wedge-type “electric field tweezers,” and the dynamic response of the surface profile is measured by an optical lever technique. Our experiments and theory indicate that when a skin layer is formed, (i) the slow relaxation process appears in the time domain and (ii) the long-persisting dip region appears in the surface profile. A parameter to quantify the difference of the surface response is proposed in this paper.

A description of how the activity of a population of neurons reflects the structure of its inputs is essential for understanding neural coding. Many studies have examined how inputs determine spiking statistics, while comparatively little is known about membrane potentials. We examine how membrane potential statistics are related to input and spiking statistics. Surprisingly, firing rates and membrane potentials are sensitive to input current modulations in distinct regimes. Additionally, the correlation between the membrane potentials of two uncoupled cells and the correlation between their spike trains reflect input correlations in distinct regimes. Our predictions are experimentally testable, provide insight into the filtering properties of neurons, and indicate that care needs to be taken when interpreting neuronal recordings that reflect a combination of subthreshold and spiking activity.

We investigate analytically the dynamics of a trapped, quasi-one-dimensional Bose-Einstein condensate subject to resonant and nonresonant periodic modulation of the transverse confinement. The dynamics of the condensate is described variationally through a set of coupled ordinary differential equations, and the period of the excited waves is determined analytically using a Mathieu-type analysis. For a modulation frequency equal to that of the radial confinement we show that the predicted period of the resonant wave is in agreement with the existing experimental results. Finally, we present a detailed comparison between the resonant waves and the Faraday waves that emerge outside of resonance.

A second order in space accurate implicit scheme for time-dependent advection-dispersion equations and a discrete fracture propagation model are employed to model solute transport in porous media. We study the impact of the fractures on mass transport and dispersion. To model flow and transport, pressure and transport equations are integrated using a finite-element, node-centered finite-volume approach. Fracture geometries are incrementally developed from a random distributions of material flaws using an adoptive geomechanical finite-element model that also produces fracture aperture distributions. This quasistatic propagation assumes a linear elastic rock matrix, and crack propagation is governed by a subcritical crack growth failure criterion. Fracture propagation, intersection, and closure are handled geometrically. The flow and transport simulations are separately conducted for a range of fracture densities that are generated by the geomechanical finite-element model. These computations show that the most influential parameters for solute transport in fractured porous media are as follows: fracture density and fracture-matrix flux ratio that is influenced by matrix permeability. Using an equivalent fracture aperture size, computed on the basis of equivalent permeability of the system, we also obtain an acceptable prediction of the macrodispersion of poorly interconnected fracture networks. The results hold for fractures at relatively low density.