Cantori are the invariant sets remaining after the destruction of KAM surfaces and create partial barriers to transport in chaotic regions. Cantori may be approximated by high-order periodic orbits; however, field line tracing methods for locating periodic orbits perform poorly in chaotic regions. To approximate cantori for continuous flow dynamics, high-order periodic orbits are determined by Lagrangian variational methods. The method is robust to chaos, converges quadratically, and the computational cost scales linearly with the periodicity length of the orbit. Minimizing-periodic orbits with periodicities in the tens of thousands, that closely approximate cantori, have been constructed.

A two-dimensional pore-network model based on invasion percolation is used to study the patterns obtained from the release of methane during the dissociation of methane hydrates (without including dissociation kinetics) caused by a sudden pressure reduction in the system below the hydrate equilibrium pressure. The concept of the critical gas saturation $S_{\mathrm{gc}}$ (volume fraction of the gas phase at the onset of bulk gas flow) is introduced to analyze gas hydrate dissociation. The effects of throat-size distribution (corresponding to off-shore oceanic sediments or on-shore sediments under permafrost), applied pressure difference across the network, and initial hydrate saturation on the resulting gas patterns and on the critical gas saturation are examined to determine the possibility of producing methane. As expected, large throat sizes or wide throat distributions, large pressure drops, and higher initial hydrate saturation act as promoters for the production of the released gas. For typical deep ocean sediments with small pore sizes and low hydrate saturation, it may be difficult to produce methane resulting from hydrate dissociation.

The backscattering of circularly polarized light pulses from an infinite uniform scattering medium is studied as a function of helicity of the incident light and size of scatterers in the medium. The approach considers a polarized short pulse of light incident on the scattering medium, and uses an analytical cumulant solution of the vector radiative transfer equation with the phase matrix obtained from the Mie theory to calculate the temporal profile of scattered polarized photons for any position and any angle of detection. The general expression for the scattered photon distribution function is an expansion in spatial cumulants up to an arbitrary high order. Truncating the expansion at the second-order cumulant, a Gaussian analytical approximate expression for the temporal profile of scattered polarized photons is obtained, whose average center position and half width are always exact. The components of scattered light copolarized and cross polarized with that of the incident light can be calculated and used for determining the degree of polarization of the scattered light. The results show that circularly polarized light of the same helicity dominates the backscattered signal when scatterer size is larger than the wavelength of light. For the scatterers smaller than the wavelength, the light of opposite helicity makes the dominant contribution to the backscattered signal. The theoretical estimates are in good agreement with our experimental results.

We report on the observation of a hexagonal superlattice state (HSS) in dielectric barrier discharge in air/argon near atmospheric pressure. It bifurcates directly from the hexagonal state by increasing the applied voltage. The correlation measurements indicate that the HSS is an interleaving of three different transient sublattices. The spatial power spectrum demonstrates that the HSS has two separate wave vectors. The big wave vector and the small wave vector belong to the harmonic mode and the subharmonic mode, respectively, and they obey the triad resonant interaction ${\stackrel{⃑}{q}}_1^s+{\stackrel{⃑}{q}}_2^s={\stackrel{⃑}{K}}_c^h$.

We model the influence of rigid inclusions on the curvature elasticity of a lipid membrane. Our focus is on conelike transmembrane inclusions that are able to induce long-range deformations in the host bilayer membrane. The elastic properties of the membrane are described in terms of curvature and tilt elasticity. The latter adds an additional degree of freedom that allows the membrane to accommodate an inclusion not only through a curvature deformation but also via changes in lipid tilt. Using a (mean-field level) cell model for homogeneously distributed inclusions in a small membrane segment of prescribed (mesoscopic-scale) spherical shape, we calculate the optimal microscopic-scale deviation of the membrane shape around the intercalated inclusions and the corresponding free energy, analytically. We show that the lipid tilt degree of freedom can lead to local softening of the inclusion-containing lipid bilayer segment. The predicted softening requires a sufficiently small value of the tilt modulus; its origin lies in the reduction of the excess membrane-inclusion interaction energy. We compare our results to the case of suppressed microscopic shape relaxation. Here, too, local softening of the membrane is possible.

By means of Monte Carlo simulations we study jamming and percolation processes upon the random sequential adsorption of dimers on binary alloys with different degrees of structural order. The substrates are equimolar mixtures that we simulate using an Ising model with conserved order parameter. After an annealing at temperature $T$ we quench the alloys to freeze the state of order of the surface at this temperature. The deposition is then performed neglecting thermal effects like surface desorption or diffusion. In this way, the annealing temperature is a continuous parameter that characterizes the adsorbing surfaces, shaping the deposition process. As the alloys undergo an order-disorder phase transition at the Onsager critical temperature $\left(T_c\right)$, the jamming and percolating properties of the set of deposited dimers are subjected to nontrivial changes, which we summarize in a density-temperature phase diagram. We find that for $T<T^{*}=1.22T_c$ the occurrence of jamming prevents the onset of percolating clusters, while percolation is possible for $T>T^{*}$. Particular attention is focused close to $T^{*}$, where the interplay between jamming and percolation restricts fluctuations, forcing exponents seemingly different from the standard percolation universality class. By analogy with a thermal transition, we study the onset of percolation using the temperature $T$ as a control parameter. We propose thermal scaling Ansätze to analyze the behavior of the percolation threshold and its thermally induced fluctuations. Also, the fractal dimension of the percolating cluster is determined. Based on these measurements and the excellent data collapse, we conclude that the universality class of standard percolation is preserved for all temperatures.

We perform extensive numerical simulations on a system of globally coupled rotors with repulsive interactions. By controlling systematically initial conditions, we determine the criterion for the emergence of bicluster motion. It is found that stable bicluster motion emerges at low temperatures, where the initial kinetic energy accounts for less than about 60% of the total energy. Also observed are collective oscillations of the potential energy and the magnetization, which are persistent. With appropriately chosen initial conditions, the system exhibits characteristic motion where biclusters keep forming and disappearing continually. It is argued that such bicluster motion is closely related to the dynamical order suggested recently.

Hollerbach and Rüdiger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this “helical” MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only), and that it might be relevant to cooler astrophysical disks, especially those around protostars, which may be quite resistive. We confirm previous results for marginal stability and calculate HMRI growth rates. We show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction along the axis. But we report other features of HMRI that make it less attractive for experiments and for resistive astrophysical disks. Large axial currents are required. More fundamentally, instability of highly resistive flow is peculiar to infinitely long or periodic cylinders: finite cylinders with insulating endcaps are shown to be stable in this limit, at least if viscosity is neglected. Also, Keplerian rotation profiles are stable in the resistive limit regardless of axial boundary conditions. Nevertheless, the addition of a toroidal field lowers thresholds for instability even in finite cylinders.

We study metastability and nucleation in a kinetic two-dimensional Ising model that is driven out of equilibrium by a small random perturbation of the usual dynamics at temperature $T$. We show that, at a mesoscopic/cluster level, a nonequilibrium potential describes in a simple way metastable states and their decay. We thus predict noise-enhanced stability of the metastable phase and resonant propagation of domain walls at low $T$. This follows from the nonlinear interplay between thermal and nonequilibrium fluctuations, which induces reentrant behavior of the surface tension as a function of $T$. Our results, which are confirmed by Monte Carlo simulations, can be also understood in terms of a Langevin equation with competing additive and multiplicative noises.

We explore the long-standing discrepancy between an experimentally observed sudden onset of the excluded-volume effect in stiff chains [Murakami et al., Macromolecules 13, 345 (1980)] and much more gradual onset seen by Monte Carlo simulations of the persistent self-avoiding walk model. We find by using various simulations and real-space renormalization methods that, while models incorporating parallel-bond inhibition enhance the excluded-volume effect, they have little effect on the abruptness of its onset. We also observe that relaxational simulations, which include reptationlike motion can, however, lead to a sudden onset much like in the experimental observation.

We present a mesoscopic hydrodynamic description of the dynamics of colloidal suspensions. We consider the system as a gas of Brownian particles suspended in a Newtonian heat bath subjected to stationary nonequilibrium conditions imposed by a velocity field. By means of a generalized Fokker-Planck equation, we obtain a set of coupled differential equations for the local diffusion current and the evolution of the total stress tensor. We find that the dynamic shear viscosity of the system contains contributions arising from the finite size of the particles.

The layer structures in bent-shaped liquid crystal dimers $m\mathrm{OAM}5\mathrm{AMO}m$ $(m=6-16)$ have been investigated by x-ray microbeam diffraction. These liquid crystal molecules have two rodlike mesogens connected with an odd-numbered alkylene spacer and form a bent shape. In these compounds it is found that the structure changes from the single $(m=6)$ to frustrated-layer structures ($m=8$, $10$, and 12) and switchable frustrated-layer structures ($m=14$ and 16) with increasing terminal chain length. An anticlinic antiferroelectric structure is suggested in the compound with $m=16$, based on the different electric-field-induced reorientation behavior from those in the other dimers.

Recent experiments found that some adhesive receptor-ligand complexes have counterintuitive catch-slip transition behaviors: the mean lifetimes of these complexes first increase (catch) with initial application of a small external force, and then decrease (slip) when the force is beyond some threshold. In this work we suggest that the forced dissociation of these complexes might be a typical rate process with dynamic disorder. The one-dimensional force modulating Agmon-Hopfield model is used to describe the transitions in the single-bond P-selectin glycoprotein ligand 1-P-selectin forced dissociation experiments, which were respectively performed in the constant force [Marshall et al., Nature (Landon) 423, 190 (2003)] and the ramping force [Evans et al., Proc. Natl. Acad. Sci. U.S.A 98, 11281 (2004)] modes. We find that, an external force can not only accelerate the bond dissociation, but also modulate the complex from the lower-energy barrier to the higher one; the catch-slip bond transition can arise from a particular energy barrier shape. The agreement between our calculation and the experimental data is satisfactory.

We address two questions that are central to understanding human motor control variability: what kind of dynamical components contribute to motor control variability (i.e., deterministic and/or random ones), and how are those components structured? To this end, we derive a stochastic order parameter equation for isometric force production from experimental data using drift-diffusion estimates. We show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity. A structural analysis reveals that the deterministic component consists of a linear control loop, while the random component involves a noise source that scales with force output. In addition, we present evidence for the existence of a subject-independent overall noise level of human isometric force production.

Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invariant hyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanical instabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce the dynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability, intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes strongly stable, the dynamics remains chaotic and no frequency locking is observed.

The von Neumann entropy for an electron in periodic, disorder, and quasiperiodic quantum small-world networks (QSWN’s) is studied numerically. For the disorder QSWN’s, the derivative of the spectrum-averaged von Neumann entropy is maximal at a certain density of shortcut links $p^{*}$, which can be as a signature of the localization-delocalization transition of electron states. The transition point $p^{*}$ is agreement with that obtained by the level statistics method. For the quasiperiodic QSWN’s, it is found that there are two regions of the potential parameter. The behaviors of electron states in different regions are similar to that of periodic and disorder QSWN’s, respectively.

We study complex networks with weights $w_{ij}$ associated with each link connecting node $i$ and $j$. The weights are chosen to be correlated with the network topology in the form found in two real world examples: (a) the worldwide airport network and (b) the E. Coli metabolic network. Here $w_{ij}\sim x_{ij}(k_i{k}_j{)}^{\alpha }$, where $k_i$ and $k_j$ are the degrees of nodes $i$ and $j$, $x_{ij}$ is a random number, and $\alpha$ represents the strength of the correlations. The case $\alpha >0$ represents correlation between weights and degree, while $\alpha <0$ represents anticorrelation and the case $\alpha =0$ reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, ${\ell }_{\mathrm{opt}}$, with the system size $N$ in strong disorder for scale-free networks for different $\alpha$. We find two different universality classes for ${\ell }_{\mathrm{opt}}$ in strong disorder depending on $\alpha$: (i) if $\alpha >0$, then for $\lambda >2$ the scaling law ${\ell }_{\mathrm{opt}}\sim N^{1∕3}$, where $\lambda$ is the power-law exponent of the degree distribution of scale-free networks, and (ii) if $\alpha \le 0$, then ${\ell }_{\mathrm{opt}}\sim N^{{\nu }_{\mathrm{opt}}}$ with ${\nu }_{\mathrm{opt}}$ identical to its value for the uncorrelated case $\alpha =0$. We calculate the robustness of correlated scale-free networks with different $\alpha$ and find the networks with $\alpha <0$ to be the most robust networks when compared to the other values of $\alpha$. We propose an analytical method to study percolation phenomena on networks with this kind of correlation, and our numerical results suggest that for scale-free networks with $\alpha <0$, the percolation threshold $p_c$ is finite for $\lambda >3$, which belongs to the same universality class as $\alpha =0$. We compare our simulation results with the real worldwide airport network, and we find good agreement.

We consider the problem of distributing a vaccine for immunizing a scale-free network against a given virus or worm. We introduce a method, based on vaccine dissemination, that seems to reflect more accurately what is expected to occur in real-world networks. Also, since the dissemination is performed using only local information, the method can be easily employed in practice. Using a random-graph framework, we analyze our method both mathematically and by means of simulations. We demonstrate its efficacy regarding the trade-off between the expected number of nodes that receive the vaccine and the network’s resulting vulnerability to develop an epidemic as the virus or worm attempts to infect one of its nodes. For some scenarios, the method is seen to render the network practically invulnerable to attacks while requiring only a small fraction of the nodes to receive the vaccine.

Exact results are derived concerning quasiregularity and diffusion of strong chaos on resonances of the sawtooth map. A chaotic ensemble of well-defined quasiregularity type (the sequence of resonances visited) is generally a fractal set whose main characteristics, the topological entropy and the Hausdorff dimension, are calculated exactly, under some conditions, using a symbolic dynamics. The effect of quasiregularity on chaotic diffusion is characterized by an infinity of diffusion coefficients, each associated with a fractal ensemble trapped in a periodic set of resonances. In some cases, these coefficients are calculated exactly and it is shown that rigorous diffusion takes place on the resonances.

We analyze pulse self-compression in femtosecond filaments, both experimentally and numerically. We experimentally demonstrate the compression of $45\mathrm{fs}$ pulses down to a duration of $7.4\mathrm{fs}$ at millijoule pulse energies. This sixfold compression in a self-generated filament does not require any means for dispersion compensation and is highly efficient. We compare our results to numerical simulations, providing a complete propagation model that accounts for full dispersion, pressure variations, Kerr nonlinearity and plasma generation in multiphoton and tunnel regimes. The equations are numerically integrated and allow for a quantitative comparison with the experiment. Our experiments and numerical simulations reveal a characteristic spectrotemporal structure of the self-compressed pulses, consisting of a compressible blue wing and an incompressible red pedestal. We explain the underlying mechanism that leads to this structure and examine the scalability of filament self-compression with respect to pulse energy and gas pressure.