We propose an order index, $\varphi$, which gives a quantitative measure of randomness and order of complete genomic sequences. It maps genomes to a number from 0 (random and of infinite length) to 1 (fully ordered) and applies regardless of sequence length. The 786 complete genomic sequences in GenBank were found to have $\varphi$ values in a very narrow range, ${\varphi }_g={0.031}_{-0.015}^{+0.028}$. We show this implies that genomes are halfway toward being completely random, or, at the “edge of chaos.” We further show that artificial “genomes” converted from literary classics have $\varphi$’s that almost exactly coincide with ${\varphi }_g$, but sequences of low information content do not. We infer that ${\varphi }_g$ represents a high information-capacity “fixed point” in sequence space, and that genomes are driven to it by the dynamics of a robust growth and evolution process. We show that a growth process characterized by random segmental duplication can robustly drive genomes to the fixed point.

We study the elastic and dynamic behavior of a self-avoiding chain confined inside a cylindrical pore using a Flory-type approach and molecular-dynamics simulations. In the Hookean regime, we find that the effective spring constant of the chain is given by $k_{\text{eff}}\sim N^{-1}{D}^{-\gamma }$, where $N$ is the number of monomers and $D$ the diameter of the pore. While the Flory approach reproduces the earlier scaling result $\gamma =1/3$, our simulations confirm a more recent numerical result $\gamma \approx 0.9$ for the computationally accessible regimes. In the absence of hydrodynamic interactions, the relaxation dynamics of a stretched-and-released chain in this regime is characterized by a global relaxation time ${\tau }_R\sim N^2{D}^{\gamma }$ with the same exponent $\gamma$ for $k_{\text{eff}}$. We also discuss how chain relaxation under confinement is influenced by hydrodynamic interactions. In the presence (or absence) of the hydrodynamic interaction, the finite-size effect observed in $k_{\text{eff}}$ is shown to persist in chain relaxation, resulting in ${\tau }_R$ markedly different from previous results.

We study the behavior of dynamical systems under time reparameterizations, which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities. We first show that time transformations are locally equivalent to metric transformations, a result that leads to a transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces. We then show that time transformations preserve the spectrum of generalized dimensions $D_q$ except for the information dimension $D_1$, which, interestingly, transforms in a nontrivial way despite previous assertions of invariance. The discontinuous behavior at $q=1$ can be used to constrain and extend the formulation of the Kaplan-Yorke conjecture.

This work is devoted to the study of steady states in two hydrodynamic traffic models. The first model to be analyzed is the Kerner-Konhäuser model and the second is called kinetic Navier-Stokes model. The study is made through an analysis in the phase space. Taking into account the analogy with the description of the one-particle motion, we can give a sensible meaning to the dynamical functions which determine the stability of the steady states. Lastly, we constructed the phase plane paths in several cases for both models.

Recurring trending feature of a particular duration or size can normally be observed in the scale-free fluctuation of economic and market data. While it contradicts the notion of being scale free, trends are generally believed to exist. From the explicit result of multiplicative cascade and empirical evidence, we show the presence of local cascades underlying the recurring trend and such characteristic is in fact an integral part rather than an aberration of the scale-free fluctuation.

Our purpose is to provide a mean-field theory for the discrete time-step susceptible-infected-recovered-susceptible (SIRS) model on uncorrelated networks with arbitrary degree distributions. The effect of network structure, time delays, and infection rate on the stability of oscillating and fixed point solutions is examined through analysis of discrete time mean-field equations. Consideration of two scenarios for disease contagion demonstrates that the manner in which contagion is transmitted from an infected individual to a contacted susceptible individual is of primary importance. In particular, the manner of contagion transmission determines how the degree distribution affects model behavior. We find excellent agreement between our theoretical results and numerical simulations on networks with large average connectivity.

Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities $c\left(\mathsf{x}\right)$ and $\tilde{c}\left(\omega ,\mathsf{x}\right)$ as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs’ wave functions $\Psi \left(x\right)$ using the inverse participation ratio and the amplitude distribution $P\mathbf{\left(}\Psi \left(x\right)\mathbf{\right)}$.

A central configuration is an arrangement of point masses in which the net gravitational accelerations are proportional to the displacements from the center of mass. Here several families of central configurations are described consisting of a large number of identical masses that occupy one or more curves. The families are found numerically. These central configurations are regular, an algebraic condition that assures their persistence in the presence of small perturbing forces such as external fields or tethering forces. Both planar and nonplanar families exist; the planar central configurations are associated with (unstable) periodic solutions to the $n$-body problem. Similar configurations are exhibited for objects having pairwise interaction proportional to $d^{-p}$ at distance $d$ for $p$ different from 2, such as point vortices.

Highly cross-linked polymer (HCP) networks are becoming increasingly important as high-performance adhesives and multifunctional composite materials. Because of their cross-linked molecular architectures, HCPs can be strong but brittle. One key goal in improving the performance of an HCP is to increase toughness without sacrificing strength. Using large scale molecular-dynamics simulation, we compare and characterize the mechanical behavior of two model HCPs under tensile deformation. In the first case, bond angles among any three connected monomers are unconstrained and in the second case we impose harmonic tetrahedral bond angle constraints. We perform a detailed microstructural analysis that establishes a unique correlation between macroscopic mechanical behavior and the microscopic structure of an HCP. While, in the unconstrained system, strain-hardening behavior is observed that is attributed to the formation of microvoids, the void growth is completely arrested in the constrained system and no strain hardening is observed. Moreover, after the initial strain-hardening phase, the unconstrained system displays the same stress-strain behavior as that of a constrained network. Strain hardening makes the unconstrained system ductile while it retains the same tensile strength as the constrained system. We suggest that bond angle flexibility of cross-linkers might be a possible means to control ductility in an HCP network at a constant cross-linker density. We have also studied the effect of temperature, strain rate, and intermonomer nonbonded interaction strength on the stress-strain behavior. Interestingly at a strong intermonomer nonbonded interaction strength, no strain hardening is observed even in the unconstrained system and fracture sets in at around 1% strain, similar to what is observed in an experimental system such as epoxy and vinyl-ester based thermosets. This indicates that strong nonbonded interactions play a key role in making an HCP strong but brittle.

Complex systems are often modeled as Boolean networks in attempts to capture their logical structure and reveal its dynamical consequences. Approximating the dynamics of continuous variables by discrete values and Boolean logic gates may, however, introduce dynamical possibilities that are not accessible to the original system. We show that large random networks of variables coupled through continuous transfer functions often fail to exhibit the complex dynamics of corresponding Boolean models in the disordered (chaotic) regime, even when each individual function appears to be a good candidate for Boolean idealization. A suitably modified Boolean theory explains the behavior of systems in which information does not propagate faithfully down certain chains of nodes. Model networks incorporating calculated or directly measured transfer functions reported in the literature on transcriptional regulation of genes are described by the modified theory.

We address the problem of how the survival of cooperation in a social system depends on the motion of the individuals. Specifically, we study a model in which prisoner’s dilemma players are allowed to move in a two-dimensional plane. Our results show that cooperation can survive in such a system provided that both the temptation to defect and the velocity at which agents move are not too high. Moreover, we show that when these conditions are fulfilled, the only asymptotic state of the system is that in which all players are cooperators. Our results might have implications for the design of cooperative strategies in motion coordination and other applications including wireless networks.

We present conditions for the local and global synchronizations in coupled-map networks using the matrix measure approach. In contrast to many existing synchronization conditions, the proposed synchronization criteria do not depend on the solution of the synchronous state and give less limitation on the network connections. Numerical simulations of the coupled quadratic maps demonstrate the potentials of our main results.

We analyze the density distribution and the adsorption of solvent hard spheres in an annular slit formed by two large solute spheres or a large solute and a wall at close distances by means of fundamental measure density-functional theory, anisotropic integral equations, and simulations. We find that the main features of the density distribution in the slit are described by an effective two-dimensional system of disks in the vicinity of a central obstacle. This has an immediate consequence for the depletion force between the solutes (or the wall and the solute) since the latter receives a strong line-tension contribution due to the adsorption of the effective disks at the circumference of the central obstacle. For large solute-solvent size ratios, the resulting depletion force has a straightforward geometrical interpretation which gives a precise “colloidal” limit for the depletion interaction. For intermediate size ratios of 5–10 and high solvent packing fractions larger than 0.4, the explicit density-functional results show a deep attractive well for the depletion potential at solute contact, possibly indicating demixing in a binary mixture at low solute and high solvent packing fraction besides the occurrence of gelation and freezing.

A continuum model was proposed to study cell spreading on a flat substrate mediated by specific interaction, long-range recruiting interaction, and the diffusion of binders. Specific interactions between the mobile receptors embedded in the cell membrane and ligands coated on the substrate surface result in cell adhesion to the substrate surface. This receptor-ligand interaction was described by a chemical reaction equation. Long-range recruiting interactions between the receptors and the substrate were simplified by a traction-separation law. The governing equations and boundary conditions were formulated for the entire process of cell spreading and solved using a finite element scheme. Parametric studies were conducted to investigate the effect of system parameters on the cell spreading kinetics. It is shown that kinetic factors play an important role in cell adhesion and three regimes, that is, the binder reaction limited regime, long-range recruiting force-driven binder recruitment limited regime, and the concentration gradient-driven diffusion limited regime, were identified.

A quantum-thermal annealing method using a cluster-flip algorithm is studied in the two-dimensional spin-glass model. The temperature $\left(T\right)$ and the transverse field $\left(\Gamma \right)$ are decreased simultaneously with the same rate along a linear path on the $T\text{−}\Gamma$ plane. We found that the additional pulse of the transverse field to the frozen local spins produces a good approximate solution with a low computational cost.

Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records $x_i$ of length $N$ that consist of a long-term correlated component $\left(1-a\right)y_i$ characterized by a correlation exponent $\gamma$, $0<\gamma <1$, and a white-noise component $a{\eta }_i$, $0\le a\le 1$. We show that the autocorrelation function $C_N\left(s\right)$ has the general form $C_N\left(s\right)=\left[C_{\infty }\left(s\right)-E_a\right]/\left(1-E_a\right)$, where $C_{\infty }\left(0\right)=1$, $C_{\infty }\left(s>0\right)=B_a{s}^{-\gamma }$, and $E_a=\left\{2B_a/\left[\left(2-\gamma \right)\left(1-\gamma \right)\right]\right\}{N}^{-\gamma }+O\left(N^{-1}\right)$. The finite-size parameter $E_a$ also occurs in related quantities, for example, in the variance ${\Delta }_N^2\left(s\right)$ of the local mean in time windows of length $s$: ${\Delta }_N^2\left(s\right)=\left[{\Delta }_{\infty }^2\left(s\right)-E_a\right]/\left(1-E_a\right)$. For purely long-term correlated data $B_0\cong \left(2-\gamma \right)\left(1-\gamma \right)/2$ yielding $E_0\cong N^{-\gamma }$, and thus $C_N\left(s\right)=\left[\frac{\left(2-\gamma \right)\left(1-\gamma \right)}{2}{s}^{-\gamma }-N^{-\gamma }\right]/\left[1-N^{-\gamma }\right]$ and ${\Delta }_N^2\left(s\right)=\left[s^{-\gamma }-N^{-\gamma }\right]/\left[1-N^{-\gamma }\right]$. We show how to estimate $E_a$ and $C_{\infty }\left(s\right)$ from a given data set and thus how to obtain accurately the exponent $\gamma$ and the amount of white noise $a$.

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy $\alpha$-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from table-top and ocean-scale fluid mixing systems to the weak quantum-classical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phase-space geometry are shown.

We study the transport properties of passive inertial particles in two-dimensional (2D) incompressible flows. Here, the particle dynamics is represented by the four-dimensional dissipative embedding map of the 2D area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter $\alpha$ and the dissipation parameter $\gamma$. The aerosol regime, where the particles are denser than the fluid, and the bubble regime, where they are less dense than the fluid, correspond to the parameter regimes $\alpha >1$ and $\alpha <1$, respectively. Earlier studies of this system show a rich phase diagram with dynamical regimes corresponding to periodic orbits, chaotic structures, and mixed regimes. We obtain the statistical characterizers of transport for this system in these dynamical regimes. These are the recurrence time statistics, the diffusion exponent, and the distribution of jump lengths. The recurrence time distribution shows a power-law tail in the dynamical regimes, where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion exponent shows behavior of three types—normal, subdiffusive, and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behavior, as well as the behavior of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes and in the behavior of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterizers of transport. We discuss the implications of our results for realistic systems.

Variational turbulence is among the few approaches providing rigorous results in turbulence. In addition, it addresses a question of direct practical interest, namely, the rate of energy dissipation. Unfortunately, only an upper bound is obtained as a larger functional space than the space of solutions to the Navier-Stokes equations is searched. Yet, in some cases, this upper bound is in good agreement with experimental results in terms of order of magnitude and power law of the imposed Reynolds number. In this paper, the variational approach to turbulence is extended to the case of dynamo action and an upper bound is obtained for the global dissipation rate (viscous and Ohmic). A simple plane Couette flow is investigated. For low magnetic Prandtl number $P_m$ fluids, the upper bound of energy dissipation is that of classical turbulence (i.e., proportional to the cubic power of the shear velocity) for magnetic Reynolds numbers below $P_m^{-1}$ and follows a steeper evolution for magnetic Reynolds numbers above $P_m^{-1}$ (i.e., proportional to the shear velocity to the power of 4) in the case of electrically insulating walls. However, the effect of wall conductance is crucial: for a given value of wall conductance, there is a value for the magnetic Reynolds number above which energy dissipation cannot be bounded. This limiting magnetic Reynolds number is inversely proportional to the square root of the conductance of the wall. Implications in terms of energy dissipation in experimental and natural dynamos are discussed.