Investigation of observability properties of nonlinear dynamical systems aims at giving a hint on how much dynamical information can be retrieved from a system using a certain measuring function. Such an investigation usually requires knowledge of the system equations. This paper addresses the challenging problem of investigating observability properties of a system only from recorded data. From previous studies it is known that phase spaces reconstructed from poor observables are characterized by local sharp pleatings, local strong squeezing of trajectories, and global inhomogeneity. A statistic is then proposed to quantify such properties of poor observability. Such a statistic was computed for a number of bench models for which observability studies had been previously performed. It was found that the statistic proposed in this paper, estimated exclusively from data, correlates generally well with observability results obtained using the system equations. It is possible to arrive at the same order of observability among the state variables using the proposed statistic even in the presence of noise with a standard deviation as high as 10% of the data. The paper includes the application of the proposed statistic to sunspot time series.

Symbolic sequences generated by coupled map lattices (CMLs) can be used to model the chaotic-like structure of genomic sequences. In this study it is shown that diffusively coupled Chebyshev maps of order 4 (corresponding to a shift of four symbols) very closely reproduce the multifractal spectrum $D_q$ of human genomic sequences for coupling constant $\alpha =0.35\pm 0.01$ if $q>0$. The presence of rare configurations causes deviations for $q<0$, which disappear if the rare event statistics of the CML is modified. Such rare configurations are known to play specific functional roles in genomic sequences serving as promoters or regulatory elements.

We investigate the escape of an ensemble of noninteracting particles inside an infinite potential box that contains a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear area-preserving mapping for the variables energy and time, leading to a mixed phase space. The chaotic sea in the phase space surrounds periodic islands and is limited by a set of invariant spanning curves. When a hole is introduced in the energy axis, the histogram of frequency for the escape of particles, which we observe to be scaling invariant, grows rapidly until it reaches a maximum and then decreases toward zero at sufficiently long times. A plot of the survival probability of a particle in the dynamics as function of time is observed to be exponential for short times, reaching a crossover time and turning to a slower-decay regime, due to sticky regions observed in the phase space.

Two different approaches are undertaken to investigate the interaction of planar shock waves with circular cylinders. Experiments are conducted in a shock-tube apparatus equipped with a schlieren-based optical system to monitor the interaction, and numerical simulations are carried out using an in-house computer code to simulate similar problems. The incident shock-wave Mach number is varied in the range 1.1–1.4. Excellent agreement is found between the simulations and the experiments in terms of shock patterns, even though the model is based on an inviscid approach. Quantitative comparisons between the experimental results for different initial conditions (shock-wave strength, cylinder diameter, and working gas) are made to find the physical parameters affecting the path of the reflected shock. An approximate universal relation is derived, which predicts the reflected-shock trajectory along the axis of symmetry as a function of the incident-shock Mach, the diameter of the cylinder, and the gas properties. This relation is valid in the vicinity of the cylinder in the range of 0.1–5 $D$, where $D$ is the cylinder diameter. It is found that the reflected shock from the cylinder evolves as in the case of a reflected-shock wave from a planar wall multiplied by a reduction factor, which depends on the incident-shock Mach number and the ratio of specific heats.

We present a study of planar physical solutions to the Lorentz-Dirac equation in a constant electromagnetic field. In this case, we reduced the Lorentz-Dirac equation to one second-order differential equation. We obtained the asymptotics of physical solutions to this equation at large proper times. It turns out that, in a crossed constant uniform electromagnetic field with vanishing invariants, a charged particle enters a universal regime at large times. We found that the ratios of momentum components that tend to constants are determined only by the external field. This effect is essentially due to a radiation reaction. There is no such effect for the Lorentz equation in this field.

We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov equation. This equation can be linearized around the homogeneous state, and a dispersion relation, which depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turns out to depend on the mode number. When this state is unstable, the growth rates are also functions of the mode number. Explicit calculations are performed for the $\alpha$-Hamiltonian mean field model with $0⩽\alpha <1$, for which the mean-field mode is always found to dominate the exponential growth. The theoretical predictions are successfully compared with numerical simulations performed on a finite lattice.

We derive a set of equations of motion for an isolated domain in an excitable reaction-diffusion system in three dimensions. In the singular limit where the interface is infinitesimally thin, the motion of the center of mass coupled with deformation is investigated near the drift bifurcation where a motionless domain becomes unstable and undergoes migration. This is an extension of our previous theory in two dimensions. We show that there are three basic motions of a domain, straight motion, rotating motion, and helical motion. The last one is a characteristic of three dimensions. The phase diagram of these three solutions is given in the parameter space of the original reaction-diffusion equations.

When subject to applied electric pulses, a lipid membrane exhibits complex responses including electrodeformation and electroporation. In this work, the electrodeformation of giant unilamellar vesicles under strong dc electric fields was investigated. Specifically, the degree of deformation was quantified as a function of the applied field strength and the electrical conductivity ratio of the fluids inside and outside of the vesicles. The vesicles were made from $L$-$\alpha$-phosphatidylcholine with diameters ranging from 14 to 30 $\mu \mathrm{m}$. Experiments were performed with field strengths ranging from 0.9 to 2.0 kV/cm, and intra-to-extra-vesicular conductivity ratios varying between 1.92 and 53.0. With these parametric configurations, the vesicles exhibited prolate elongations along the direction of the electric field. The degree of deformation was, in general, significant. In some cases, the aspect ratio of a deformed vesicle exceeded 10, representing a strong-deformation regime previously not explored. The aspect ratio scaled quadratically with the field strength, and increased asymptotically to a maximum value at high conductivity ratios. Appreciable area and volumetric changes were observed both during and after pulsation, indicating the concurrence of electroporation. A theoretical model is developed to predict these large deformations in the strongly permeabilized limit, and the results are compared with the experimental data. Both agreements and discrepancies are found, and the model limitations and possible extensions are discussed.

A spectroscopic method is discussed to measure core and compressed-shell conditions in thick-wall plastic-shell implosions filled with deuterium and a tracer amount of argon. Simultaneous observation over a broad photon energy range of the argon line emission and the attenuation and self-emission effects of the compressed shell confining the core yields enough information to extract average temperature and density conditions in both core and compressed shell. The spectroscopic analysis also provides an estimate of the target areal density which is an important characteristic of inertial confinement fusion implosions.

Null transmission ellipsometry was employed to study the temperature evolution of the helical structure of the smectic-$C$${}_{\alpha }^{*}$ phase. Free-standing films with thickness ranging from 31 to more than 400 layers were prepared and studied. The experimental results show a reduced twisting power in thin films. A simple model was constructed to explain the results. Surface effects were found to be the key reason for this phenomenon. Our findings are consistent with the theoretical studies of helically ordered magnetic films.

Using the generic example of Ising spins on scale-free networks, we demonstrate that degree-degree correlations can induce a large number of thermodynamically stable states in networks that otherwise exhibit only the two completely ordered states. The additional stable states are related to the layered network structure. As one increases the temperature, a cascade of first-order phase transitions is found, at which some layers of the network become disordered, while others remain ordered. Negative degree-degree correlations are found to stabilize ordered layers against thermal fluctuations. Positively correlated networks can exhibit an infinite number of ground states and phase transitions, while in negatively correlated networks both numbers are finite.

We study an ideal-gas-like model where the particles exchange energy stochastically, through energy-conserving scattering processes, which take place if and only if at least one of the two particles has energy below a certain energy threshold (interactions are initiated by such low-energy particles). This model has an intriguing phase transition in the sense that there is a critical value of the energy threshold below which the number of particles in the steady state goes to zero, and above which the average number of particles in the steady state is nonzero. This phase transition is associated with standard features like “critical slowing down” and nontrivial values of some critical exponents characterizing the variation of thermodynamic quantities near the threshold energy. The features are exhibited not only in the mean-field version but also in the lattice versions.

The temporal behavior of the mean-square displacement and the velocity autocorrelation function of a particle subjected to a periodic force in a harmonic potential well is investigated for viscoelastic media using the generalized Langevin equation. The interaction with fluctuations of environmental parameters is modeled by a multiplicative white noise, by an internal Mittag-Leffler noise with a finite memory time, and by an additive external noise. It is shown that the presence of a multiplicative noise has a profound effect on the behavior of the autocorrelation functions. Particularly, for correlation functions the model predicts a crossover between two different asymptotic power-law regimes. Moreover, a dependence of the correlation function on the frequency of the external periodic forcing occurs that gives a simple criterion to discern the multiplicative noise in future experiments. It is established that additive external and internal noises cause qualitatively different dependences of the autocorrelation functions on the external forcing and also on the time lag. The influence of the memory time of the internal noise on the dynamics of the system is also discussed.

In a recent paper we exposed a filled elastomer to controlled radiation dosages and explored changes in its cross-link density and molecular weight distribution between network junctions [A. Maiti et al., Phys. Rev. E 83, 031802 (2011)]. Here we report mechanical response measurements when the material is exposed to radiation while being under finite nonzero strain. We observe interesting hysteretic behavior and material softening representative of the Mullins effect, and materials hardening due to radiation. The net magnitude of the elastic modulus depends upon the radiation dosage, strain level, and strain-cycling history of the material. Using the framework of Tobolsky's two-stage independent network theory we develop a model that can quantitatively interpret the observed elastic modulus and its radiation and strain dependence.

We measure the three-phase oil relative permeability $k_{\mathit{ro}}$ by conducting unsteady-state drainage experiments in a 0.8 $m$ water-wet sand pack. We find that when starting from capillary-trapped oil, $k_{\mathit{ro}}$ shows a strong dependence on both the flow of water and the water saturation and a weak dependence on oil saturation, contrary to most models. The observed flow coupling between water and oil is stronger in three-phase flow than two-phase flow, and cannot be observed in steady-state measurements. The results suggest that the oil is transported through moving gas-oil-water interfaces (form drag) or momentum transport across stationary interfaces (friction drag). We present a simple model of friction drag which compares favorably to the experimental data.

An algorithm for handling hard-wall interactions in simulations of driven diffusive particle motion is proposed. It exploits an exact expression for the one-dimensional transition probability in the presence of a hard (reflecting) wall and therefore is numerically exact in the sense that it does not introduce any additional approximation beyond the usual discretization procedures. Studying two standard situations from soft matter systems, its performance is compared to the heuristic approaches used in the literature.

The impact of an inhomogeneous arrangement of nodes in space on a network organization cannot be neglected in most real-world scale-free networks. Here we propose a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to $\gamma =2$ regardless of the fractal dimension. We show that this phenomenon is related to the transition from a noncompact to compact phase of the network and that this transition accompanies a drastic change of the network efficiency. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.

The principal eigenvalue $\lambda$ of a network’s adjacency matrix often determines dynamics on the network (e.g., in synchronization and spreading processes) and some of its structural properties (e.g., robustness against failure or attack) and is therefore a good indicator for how “strongly” a network is connected. We study how $\lambda$ is modified by the addition of a module, or community, which has broad applications, ranging from those involving a single modification (e.g., introduction of a drug into a biological process) to those involving repeated additions (e.g., power-grid and transit development). We describe how to optimally connect the module to the network to either maximize or minimize the shift in $\lambda$, noting several applications of directing dynamics on networks.

We present an analysis of the quasistationary (QS) state of the contact process (CP) on annealed scale-free networks using a mapping of the CP dynamics in a one-step process and analyzing numerically and analytically the corresponding master equation. The relevant QS quantities determined via the master equation exhibit an excellent agreement with direct QS stochastic simulations of the CP. The high accuracy of the resulting data allows a probe of the strong corrections to scaling present in both the critical and supercritical regions, corrections that mask the correct finite-size scaling obtained analytically by applying an exact heterogeneous mean-field approach. Our results represent a promising starting point for a deeper understanding of the contact process and absorbing phase transitions on real (quenched) complex networks.

In a recent study [S. Das, J. H. Snoeijer, and D. Lohse, Phys. Rev. E 82, 056310 (2010)], we provided quantitative demonstration of the conjecture [W. A. Ducker, Langmuir 25, 8907 (2009)] that the presence of impurities at the surface layer (or the air-water interface) of surface nanobubbles can substantially lower the gas-side contact angle and the Laplace pressure of the nanobubbles. Through an analytical model for any general air-water interface without nonideality effects, we showed that a large concentration of soluble impurities at the air-water interface of the nanobubbles ensures significantly small contact angles (matching well with the experimental results) and Laplace pressure (though large enough to forbid stability). In this paper this general model is extended to incorporate the effect of nonidealities at the air-water interface in impurity-induced alteration of surface nanobubble properties. Such nonideality effects arise from finite enthalpy or entropy of mixing or finite ionic interactions of the impurity molecules at the nanobubble air-water interface and ensure significant lowering of the nanobubble contact angle and Laplace pressure even at relatively small impurity coverage. In fact for impurity molecules that show enhanced tendency to get adsorbed at the nanobubble air-water interface from the bulk phase, impurity-induced lowering of the nanobubble contact angle is witnessed for extremely small bulk concentration. Surface nanobubble experiments being typically performed in an ultraclean environment, the bulk concentration of impurities is inevitably very small, and in this light the present calculations can be viewed as a satisfactory explanation of the conjecture that impurities, even in trace concentration, have significant impact on surface nanobubbles.