The effective one-dimensional description of diffusion in two-dimensional channels of varying cross section is revisited. The effective diffusion coefficient $D\left(x\right)$, extending Fick-Jacobs equation, depending on the longitudinal coordinate $x$, is derived here without use of scaling of the transverse coordinates. The result of the presented method is an integral formula for $D\left(x\right)$, calculating its value at $x$ as an integral of contributions from the neighboring positions $x^{\prime }$ depending on $h\left(x^{\prime }\right)$, a function shaping the channel. Unlike the standard formulas based on the scaling, the new proposed formula also describes $D\left(x\right)$ correctly near the cusps, or in wider channels.

Act-and-wait modification of a time-delayed feedback control (TDFC) algorithm is proposed to stabilize unstable periodic orbits in nonautonomous dynamical systems. Due to periodical switching on and off the control perturbation, an infinite-dimensional function space of the TDFC system is reduced to the finite-dimensional state space. As a result the number of Floquet exponents defining the stability of the controlled orbit remains the same as for the control-free system. The values of these exponents can be effectively manipulated by the variation of control parameters. We demonstrate the advantages of the modification for the chaotic nonautonomous Duffing oscillator with diagonal and nondiagonal control matrices. In both cases very deep minima of the spectral abscissa of Floquet exponents have been attained. The advantage of the modification is particularly remarkable for the nondiagonal coupling; in this case the conventional TDFC fails, whereas the modified version works.

Our aim is to unveil how resonances of parametric systems are affected when symmetry is broken. We showed numerically and experimentally that odd resonances indeed come about when the pendulum is excited along a tilted direction. Applying the Melnikov subharmonic function, we not only determined analytically the loci of saddle-node bifurcations delimiting resonance regions in parameter space but also explained these observations by demonstrating that, under the Melnikov method point of view, odd resonances arise due to an extra torque that appears in the asymmetric case.

We study synchronization in heterogeneous FitzHugh-Nagumo networks. It is well known that heterogeneities in the nodes hinder synchronization when becoming too large. Here we develop a controller to counteract the impact of these heterogeneities. We first analyze the stability of the equilibrium point in a ring network of heterogeneous nodes. We then derive a sufficient condition for synchronization in the absence of control. Based on these results we derive the controller providing synchronization for parameter values where synchronization without control is absent. We demonstrate our results in networks with different topologies. Particular attention is given to hierarchical (fractal) topologies, which are relevant for the architecture of the brain.

Global synchronization and partial synchronization are the two distinctive forms of synchronization in coupled oscillators and have been well studied in recent decades. Recent attention on synchronization is focused on the chimera state (CS) and explosive synchronization (ES), but little attention has been paid to their relationship. Here we study this topic by presenting a model to bridge these two phenomena, which consists of two groups of coupled oscillators, and its coupling strength is adaptively controlled by a local order parameter. We find that this model displays either CS or ES in two limits. In between the two limits, this model exhibits both CS and ES, where CS can be observed for a fixed coupling strength and ES appears when the coupling is increased adiabatically. Moreover, we show both theoretically and numerically that there are a variety of CS basin patterns for the case of identical oscillators, depending on the distributions of both the initial order parameters and the initial average phases. This model suggests a way to easily observe CS, in contrast to other models having some (weak or strong) dependence on initial conditions.

Analysis of short-pulse propagation in positive dispersion media, e.g., in optical fibers and in shallow water, requires assorted high-order derivative terms. We present an infinite-order “dark” hierarchy of equations, starting from the basic defocusing nonlinear Schrödinger equation. We present generalized soliton solutions, plane-wave solutions, and periodic solutions of all orders. We find that “even”-order equations in the set affect phase and “stretching factors” in the solutions, while “odd”-order equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are complex. There are various applications in optics and water waves.

We present a characterization of short-term stability of Kauffman's $NK$ (random) Boolean networks under arbitrary distributions of transfer functions. Given such a Boolean network where each transfer function is drawn from the same distribution, we present a formula that determines whether short-term chaos (damage spreading) will happen. Our main technical tool which enables the formal proof of this formula is the Fourier analysis of Boolean functions, which describes such functions as multilinear polynomials over the inputs. Numerical simulations on mixtures of threshold functions and nested canalyzing functions demonstrate the formula's correctness.

Most random graph models are locally tree-like—do not contain short cycles—rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of HCM, we observe two previously undiscovered power-law relations: between the number of edges inside a community and the community sizes, and between the number of edges going out of a community and the community sizes. We also relate the power-law exponent $\tau$ of the degree distribution with the power-law exponent of the community-size distribution $\gamma$. In the case of extremely dense communities (e.g., complete graphs), this relation takes the simple form $\tau =\gamma -1$.

We analyze the dynamics of a discharging electrolytic cell comprised of a binary symmetric electrolyte between two planar, parallel blocking electrodes. When a voltage is initially applied, ions in the electrolyte migrate towards the electrodes, forming electrical double layers. After the system reaches steady state and the external current decays to zero, the applied voltage is switched off and the cell discharges, with the ions eventually returning to a uniform spatial concentration. At voltages on the order of the thermal voltage $V_T=k_{B}T/q\simeq 25$ mV, where $k_B$ is Boltzmann's constant, $T$ is temperature, and $q$ is the charge of a proton, experiments on surfactant-doped nonpolar fluids observe that the temporal evolution of the external current during charging and discharging is not symmetric [V. Novotny and M. A. Hopper, J. Electrochem. Soc. 126, 925 (1979); P. Kornilovitch and Y. Jeon, J. Appl. Phys. 109, 064509 (2011)]. In fact, at sufficiently large voltages (several $V_T$), the current during discharging is no longer monotonic: it displays a “reverse peak” before decaying in magnitude to zero. We analyze the dynamics of discharging by solving the Poisson-Nernst-Planck equations governing ion transport via asymptotic and numerical techniques in three regimes. First, in the “linear regime” when the applied voltage $V$ is formally much less than $V_T$, the charging and discharging currents are antisymmetric in time; however, the potential and charge density profiles during charging and discharging are asymmetric. The current evolution is on the $RC$ timescale of the cell, ${\lambda }_{D}L/D$, where $L$ is the width of the cell, $D$ is the diffusivity of ions, and ${\lambda }_D$ is the Debye length. Second, in the (experimentally relevant) thin-double-layer limit $\epsilon ={\lambda }_D/L\ll 1$, there is a “weakly nonlinear” regime defined by $V_T\lesssim V\lesssim V_{T}ln(1/\epsilon )$, where the bulk salt concentration is uniform; thus the $RC$ timescale of the evolution of the current magnitude persists. However, nonlinear, voltage-dependent, capacitance of the double layer is responsible for a break in temporal antisymmetry of the charging and discharging currents. Third, the reverse peak in the discharging current develops in a “strongly nonlinear” regime $V\gtrsim V_{T}ln(1/\epsilon )$, driven by neutral salt adsorption into the double layers and consequent bulk depletion during charging. The strongly nonlinear regime features current evolution over three timescales. The current decays in magnitude on the double layer relaxation timescale, ${\lambda }_D^2/D$; then grows exponentially in time towards the reverse peak on the diffusion timescale, $L^2/D$, indicating that the reverse peak is the results of fast diffusion of ions from the double layer layer to the bulk. Following the reverse peak, the current decays exponentially to zero on the $RC$ timescale. Notably, the current at the reverse peak and the time of the reverse peak saturate at large voltages $V\gg V_{T}ln(1/\epsilon )$. We provide semi-analytic expressions for the saturated reverse peak time and current, which can be used to infer charge carrier diffusivity and concentration from experiments.

We study the dynamics of individual polystyrene ellipsoids of different aspect ratios trapped at the air-water interface. Using particle tracking and in situ vertical scanning interferometry techniques we are able to measure translational drags and the protrusion in air of the ellipsoids. We report that translational drags on the ellipsoid are unexpectedly enhanced: despite the fact that a noticeable part of the ellipsoid is in air, drags are found larger than the bulk one in water.

Motivated by observations of heterogeneous domain structure on the surface of cells, we consider a minimal model to describe the dynamics of phase separation on the surface of a spherical particle. Finite-size effects on the curved particle surface lead to the formation of long-lived, metastable states for which the density is distributed in patches over the particle surface. We study the time evolution and stability of these states as a function of both the particle size and the thermodynamic parameters. Finally, by connecting our findings with studies of patchy particles, we consider the implications for self-assembly in many-particle systems.

Many biological systems fold thin sheets of lipid membrane into complex three-dimensional structures. This microscopic origami is often mediated by the adsorption and self-assembly of proteins on a membrane. As a model system to study adsorption-mediated interactions, we study the collective behavior of micrometric particles adhered to a lipid vesicle. We estimate the colloidal interactions using a maximum likelihood analysis of particle trajectories. When the particles are highly wrapped by a tense membrane, we observe strong long-range attractions with a typical binding energy of $150k_{B}T$ and significant forces extending a few microns.

We derive a mathematical model of a nematic electrolyte based on a variational formulation of nematodynamics. We verify the model by comparing its predictions to the results of the experiments on the substrate-controlled liquid-crystal-enabled electrokinetics. In the experiments, a nematic liquid crystal confined to a thin planar cell with surface-patterned anchoring conditions exhibits electro-osmotic flows along the “guiding rails” imposed by the spatially varying director. Extending our previous work, we consider a general setup which incorporates dielectric anisotropy of the liquid-crystalline matrix and the full set of nematic viscosities.

The time evolution of the contact force structure of a consolidated granular material subjected to a constant stress is monitored using the coda wave interferometry method. In addition, the nature of the aging and rejuvenation processes are investigated. These processes are interpreted in terms of affine and nonaffine structural path deformations. During the later stages of creep, the rearrangements of subgrains are so small that they only produce affine deformations in the contact paths, without any significant changes in the structural configuration. As a result, the strain path distribution follows the macroscopic strain. Conversely, in the presence of ultrasonic perturbations, the nonaffine grain buckling mechanism dominates, producing relatively drastic changes in the structural configuration accompanied by path deformations of the order of the grain size. This plastic mechanism induces material rejuvenation that is observed macroscopically as an ultrasonically accelerated creep.

Phase-field simulations are performed to explore the thermal solidification of a pure melt in three-dimensional capillaries. Motivated by our previous work for isotropic or slightly anisotropic materials, we focus here on the more general case of anisotropic materials. Different channel cross sections are compared (square, hexagonal, circular) to reveal the influence of geometry and the effects of a competition between the crystal and the channel symmetries. In particular, a compass effect toward growth directions favored by the surface energy is identified. At given undercooling and anisotropy, the simulations generally show the coexistence of several growth modes. The relative stability of these growth modes is tested by submitting them to a strong spatiotemporal noise for a short time, which reveals a subtle hierarchy between them. Similarities and differences with experimental growth modes in confined geometry are discussed qualitatively.

Accurate virial coefficients $B_N(\lambda ,\varepsilon )$ (where $\varepsilon$ is the well depth) for the three-dimensional square-well and square-step potentials are calculated for orders $N=5–9$ and well widths $\lambda =1.1-2.0$ using a very fast recursive method. The efficiency of the algorithm is enhanced significantly by exploiting permutation symmetry and by storing integrands for reuse during the calculation. For $N=9$ the storage requirements become sufficiently large that a parallel algorithm is developed. The methodology is general and is applicable to other discrete potentials. The computed coefficients are precise even near the critical temperature, and thus open up possibilities for analysis of criticality of the system, which is currently not accessible by any other means.

The probability distribution for the free energy of directed polymers in random media (DPRM) with uncorrelated noise in $d=1+1$ dimensions satisfies the Tracy-Widom distribution. We inquire if and how this universal distribution is modified in the presence of spatially correlated noise. The width of the distribution scales as the DPRM length to an exponent $\beta$, in good (but not full) agreement with previous renormalization group and numerical results. The scaled probability is well described by the Tracy-Widom form for uncorrelated noise, but becomes symmetric with increasing correlation exponent. We thus find a class of distributions that continuously interpolates between Tracy-Widom and Gaussian forms.

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.

Kramers' rate for the passage of trajectories $X\left(t\right)$ over an energy barrier due to thermal or other fluctuations is usually associated with additive noise. We present a generalization of Kramers' rate for systems with multiplicative noise. We show that the expression commonly used in the literature for multiplicative noise is not correct, and we present results of numerical integrations of the Langevin equation for $dX\left(t\right)/dt$ evolving in a quartic bistable potential which corroborate our claim.