For a fixed number $N$ of nodes, the number of links $L$ in the line graph $H(N,L)$ can only appear in consecutive intervals, called a band of $L$. We prove that some consecutive integers can never represent the number of links $L$ in $H(N,L)$, and they are called a bandgap of $L$. We give the exact expressions of bands and bandgaps of $L$. We propose a model which can randomly generate simple graphs which are line graphs of other simple graphs. The essence of our model is to merge step by step a pair of nodes in cliques, which we use to construct line graphs. Obeying necessary rules to ensure that the resulting graphs are line graphs, two nodes to be merged are randomly chosen at each step. If the cliques are all of the same size, the assortativity of the line graphs in each step are close to 0, and the assortativity of the corresponding root graphs increases linearly from $-1$ to 0 with the steps of the nodal merging process. If we dope the constructing elements of the line graphs—the cliques of the same size—with a relatively smaller number of cliques of different size, the characteristics of the assortativity of the line graphs is completely altered. We also generate line graphs with the cliques whose sizes follow a binomial distribution. The corresponding root graphs, with binomial degree distributions, zero assortativity, and semicircle eigenvalue distributions, are equivalent to Erdős-Rényi random graphs.

The relaxation dynamics in mixed chaotic systems are believed to decay algebraically with a universal decay exponent that emerges from the hierarchical structure of the phase space. Numerical studies, however, yield a variety of values for this exponent. In order to reconcile these results, we consider an ensemble of mixed chaotic systems approximated by rate equations and analyze the fluctuations in the distribution of Poincaré recurrence times. Our analysis shows that the behavior of these fluctuations, as a function of time, implies a very slow convergence of the decay exponent of the relaxation.

Chaotic data generated by a three-dimensional dynamical system can be embedded into ${\mathbb{R}}^3$ in a number of inequivalent ways. However, when lifted into ${\mathbb{R}}^5$ they all become equivalent, indicating that they all belong to a single universality class sharing a common chaos-generating mechanism. We present a complete invariant determining this universality class and distinguishing attractors generated by distinct mechanisms. This invariant is easily computable from an appropriately “dressed” return map of any particular three-dimensional embedding.

An intrinsic localized mode (ILM) represents a localized vibrational excitation in a nonlinear lattice. Such a mode will stay in resonance as the driver frequency is changed adiabatically until a bifurcation point is reached, at which point the ILM switches and disappears. The dynamics behind switching in such a many body system is examined here through experimental measurements and numerical simulations. Linear response spectra of a driven micromechanical array containing an ILM were measured in the frequency region between two fundamentally different kinds of bifurcation points that separate the large amplitude ILM state from the two low amplitude vibrational states. Just as a natural frequency can be associated with a driven harmonic oscillator, a similar natural frequency has been found for a driven ILM via the beat frequency between it and a weak, tunable probe. This finding has been confirmed using numerical simulations. The behavior of this nonlinear natural frequency plays important but different roles as the two bifurcation points are approached. At the upper transition its frequency coalesces with the driver and the resulting bifurcation is very similar to the saddle-node bifurcation of a single driven Duffing oscillator, which is treated in an Appendix. The lower transition occurs when the four-wave mixing partner of the natural frequency of the ILM intersects the topmost extended band mode of the same symmetry. The properties of linear local modes associated with the driven ILM are also identified experimentally for the first time and numerically but play no role in these transitions.

Multifractal detrended fluctuation analysis (MF-DFA) is a generalization of the conventional multifractal analysis. It is extended from the detrended fluctuation analysis (DFA) which is developed for the purpose of detecting long-range correlation and fractal property in stationary and nonstationary time series. The MF-DFA and some corresponding relationships of the exponents have been subsequently extended to the two-dimensional space. We reexamine two extended relationships in this study and demonstrate that: (i) The invalidity of the relationship $h\left(q\right)\equiv H$ for two-dimensional fractional Brownian motion, and $h(q=2)\equiv H$ between the Hurst exponent $H$ and the generalized Hurst exponent $h\left(q\right)$ in the two-dimensional case. Two more logical relationships are proposed instead as $h(q=2)=H$ for the stationary surface and $h(q=2)=H+2$ for the nonstationary signal. (ii) The invalidity of the expression $\tau \left(q\right)=qh\left(q\right)-D_f$ stipulating the relationship between the standard partition-function-based multifractal exponent $\tau \left(q\right)$ and the generalized Hurst exponent $h\left(q\right)$ in the two-dimensional case. Reasons for its invalidity are given from two perspectives.

The analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of $N$ stochastic excitable units each is performed by studying an approximate system, obtained by replacing each population with the corresponding mean-field model. In the exact system, one has the units within an ensemble communicating via the time-delayed linear couplings, whereas the interensemble terms involve the nonlinear time-delayed interaction mediated by the appropriate global variables. The aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system, governed by a set of $4N$ stochastic delay-differential equations for the microscopic dynamics, can accurately be reproduced by a flow containing just four deterministic delay-differential equations which describe the evolution of the mean-field based variables. In particular, the considered issues include determining the parameter domains where the stationary state is stable, the scenarios for the onset, and the time-delay induced suppression of the collective mode, as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state. We show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations, like those with symmetrical or asymmetrical interpopulation couplings.

Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real physical one. A generic family of stochastic processes describing noise-driven intermittency, where the stationary density distribution coincides exactly with the one resulted from entropy maximization, is presented.

We study by computer simulation a recently introduced generalized model of self-interacting self-avoiding trails on the square lattice that distinguishes two topologically different types of self-interaction: namely, crossings where the trail passes across itself and collisions where the lattice path visits the same site without crossing. This model generalizes the canonical interacting self-avoiding trail model of polymer collapse, which has a strongly divergent specific heat at its transition point. We confirm the recent prediction that the asymmetry does not affect the universality class for a range of asymmetry. Certainly, where the weighting of collisions outweighs that of crossings this is well supported numerically. When crossings are weighted heavily relative to collisions, the collapse transition reverts to the canonical $\theta$-point-like behavior found in interacting self-avoiding walks.

Recent experimental and theoretical studies suggest that crystallization and glass-like solidification are useful analogies for understanding cell ordering in confluent biological tissues. It remains unexplored how cellular ordering contributes to pattern formation during morphogenesis. With a computational model we show that a system of elongated, cohering biological cells can get dynamically arrested in a network pattern. Our model provides an explanation for the formation of cellular networks in culture systems that exclude intercellular interaction via chemotaxis or mechanical traction.

Gene networks exhibiting oscillatory dynamics are widespread in biology. The minimal regulatory designs giving rise to oscillations have been implemented synthetically and studied by mathematical modeling. However, most of the available analyses generally neglect the coupling of regulatory circuits with the cellular “chassis” in which the circuits are embedded. For example, the intracellular macromolecular composition of fast-growing bacteria changes with growth rate. As a consequence, important parameters of gene expression, such as ribosome concentration or cell volume, are growth-rate dependent, ultimately coupling the dynamics of genetic circuits with cell physiology. This work addresses the effects of growth rate on the dynamics of a paradigmatic example of genetic oscillator, the repressilator. Making use of empirical growth-rate dependencies of parameters in bacteria, we show that the repressilator dynamics can switch between oscillations and convergence to a fixed point depending on the cellular state of growth, and thus on the nutrients it is fed. The physical support of the circuit (type of plasmid or gene positions on the chromosome) also plays an important role in determining the oscillation stability and the growth-rate dependence of period and amplitude. This analysis has potential application in the field of synthetic biology, and suggests that the coupling between endogenous genetic oscillators and cell physiology can have substantial consequences for their functionality.

The cell cycle is a tightly controlled process, yet it shows marked differences across species. Which of its structural features follow solely from the ability to control gene expression? We tackle this question in silico by examining the ensemble of all regulatory networks which satisfy the constraint of producing a given sequence of gene expressions. We focus on three cell cycle profiles coming from baker's yeast, fission yeast, and mammals. First, we show that the networks in each of the ensembles use just a few interactions that are repeatedly reused as building blocks. Second, we find an enrichment in network motifs that is similar in the two yeast cell cycle systems investigated. These motifs do not have autonomous functions, yet they reveal a regulatory logic for cell cycling based on a feed-forward cascade of activating interactions.

Traffic flow complexity comes from the car-following and lane-changing behavior. Based on empirical data for individual vehicle speeds and time headways measured on a single-lane highway section, we have studied the traffic flow properties induced by pure car-following behavior. We have found that a spontaneous sudden drop in velocity could happen in a platoon of vehicles when the velocity of the leading vehicle is quite high (∼70 km/h). In contrast, when the velocity of the leading vehicle in a platoon slows down, such a spontaneous sudden drop of velocity has not been observed. Our finding indicates that traffic breakdown on a single-lane road might be a phase transition from free flow to synchronized flow ($F\to S$ transition). We have found that the flow rate within the emergent synchronized flow can be either smaller or larger than the flow rate in the free flow within which the synchronized flow propagates. Our empirical findings support Kerner's three-phase theory in which traffic breakdown is associated with an $F\to S$ transition.

We compare the 2D and 3D classical fractal scattering dynamics of Cl and HO for energies just above dissociation of the HOCl molecule, using a realistic potential energy surface for the HOCl molecule and techniques developed to analyze 3D chaotic scattering processes. For parameter regimes where the HO dimer initially has small vibrational energy, only small intervals of initial conditions show fractal scattering behavior and the scattering process is well described by a 2D model. For parameter regimes where the HO dimer initially has large vibrational energy, the scattering process is fully 3D and is dominated by fractal behavior.

The present paper addresses the electromagnetic field generated by a point charge or a small charged particle bunch moving with constant velocity in a cold magnetized plasma, along the external magnetic field. Attention is focused on the case of ultrarelativistic motion. The field surrounding the point charge is investigated both analytically and numerically. In the analytical study, we obtain rigorous decomposition of the field into quasistatic and wave components. Beating behavior in the far-field zone and harmonic behavior in the vicinity of the charge trajectory are found using suitable approximate approaches. The transverse component of the electric field exhibits a strong (inversely proportional) singularity on the charge trajectory, while the longitudinal components of both the electric and magnetic fields exhibit a weaker (logarithmic) singularity. An efficient numerical approach is developed to calculate the field for arbitrary parameters. An efficient algorithm for calculating the fields of small bunches with different forms is also presented, using a thin charged disk and a charged cylinder as representative examples.

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of ${6144}^3$ points and three different configurations on grids of ${4096}^3$ points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between $t=2.33$ and $t=2.70.$ More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

A simple photon cavity consisting of three flat mirrors is considered, which provides a model for a photon neutralizer of negative ion beams to be used in a neutral beam injector for plasma heating in fusion devices. Treating the system as an open mathematical billiard, the mechanism of photon scattering on the sharp boundary between two mirrors is described. Assuming a balance between the continuous injection of the photons into the device and their loss through its open ends, the stationary angular distribution of the photons in the trap is computed in closed form and an estimate for the confinement time of the photons in the neutralizer is provided.

We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.

Anchoring transition (ATr), resulting in the orientational change of liquid crystal (LC) molecules at surfaces with temperature variation, is shown to be essentially affected by the application of an electric field ($E$ field). By combining defocusing polarizing microscopy with retardation measurement, a detailed investigation of the molecular orientational change in the vicinity of ATr during a cooling process under an $E$ field was performed. Two important conclusions can be made from our observations: (i) the ATr temperature exhibits a large downshift in the presence of an $E$ field; (ii) the orientational change is always discontinuous from a planar ($P$) state to a homeotropic ($H$) state. If the applied $E$ field is below the Freedericksz transition, $P$ and $H$ states are uniform, whereas the surface $H$ state is influenced by the tilt in the bulk due to negative dielectric anisotropy, resulting in a bent-splay deformed state.

Psychological and physiological considerations entail that formation and functionality of neuronal cell assemblies depend upon synchronized repeated activation such as zero-lag synchronization. Several mechanisms for the emergence of this phenomenon have been suggested, including the global network quantity, the greatest common divisor of neuronal circuit delay loops. However, they require strict biological prerequisites such as precisely matched delays and connectivity, and synchronization is represented as a stationary mode of activity instead of a transient phenomenon. Here we show that the unavoidable increase in neuronal response latency to ongoing stimulation serves as a nonuniform gradual stretching of neuronal circuit delay loops. This apparent nuisance is revealed to be an essential mechanism in various types of neuronal time controllers, where synchronization emerges as a transient phenomenon and without predefined precisely matched synaptic delays. These findings are described in an experimental procedure where conditioned stimulations were enforced on a circuit of neurons embedded within a large-scale network of cortical cells in vitro, and are corroborated and extended by simulations of circuits composed of Hodgkin-Huxley neurons with time-dependent latencies. These findings announce a cortical time scale for time controllers based on tens of microseconds stretching of neuronal circuit delay loops per spike. They call for a reexamination of the role of the temporal periodic mode in brain functionality using advanced in vitro and in vivo experiments.