Chaotic synchronization for a pair of electrochemical oscillators is studied experimentally. The underlying bidirectional coupling between the two oscillators is achieved by immersing the two anodes in a common electrolytic solution. The horizontal distance between these two electrodes determines the strength of the coupling constant. On monotonically decreasing the distance between the two anodes, different domains of chaotic synchronization, namely, no, phase, lag, and complete synchronization, are identified. Furthermore, dynamics from the different transition intervals are also characterized.

We show how the irreducible memory function can be obtained in a rather straightforward way, and that it can be expressed in terms of two contributions representing two parallel decay channels. This representation should be useful for treating systems with a slow time dependence and where eventually some internal degrees of freedom enters in the relaxation process, and cuts off an underlying ideal ergodic to nonergodic transition. We also show how the irreducible memory function under certain mild conditions defines a regenerative stochastic process, or a two level stochastic system. This leads to a picture with dynamical heterogeneities, where the statistical properties asymptotically are ruled by limit processes. This can explain the universal behavior observed in many glass-forming systems.

We present analytical results for the finite-size scaling in $d$-dimensional $O\left(N\right)$ systems with strong anisotropy where the critical exponents (e.g., ${\nu }_{\Vert }$ and ${\nu }_{\perp }$) depend on the direction. Prominent examples are systems with long-range interactions, decaying with the interparticle distance $r$ as $r^{-d-\sigma }$ with different exponents $\sigma$ in corresponding spatial directions, systems with space-“time” anisotropy near a quantum critical point, and systems with Lifshitz points. The anisotropic properties involve also the geometry of the systems. We consider $O\left(N\right)$ systems in the $N\to \infty$ limit, confined to a $d$-dimensional layer with geometry $L^m\times {\infty }^n;m+n=d$ and periodic boundary conditions across the finite $m$ dimensions. The arising difficulties are avoided using a technique of calculations based on the analytical properties of the generalized Mittag-Leffler functions.

We study resonances of chaotic map dynamics. We use the calculus of variations to determine the additive forcing function that induces the largest response. We find that resonant forcing functions complement the separation of nearby trajectories, in that the product of the displacement of nearby trajectories and the resonant forcing is a conserved quantity. As a consequence, the resonant function will have the same periodicity as the displacement dynamics, and if the displacement dynamics is irregular, then the resonant forcing function will be irregular as well. Furthermore, we show that resonant forcing functions of chaotic systems decrease exponentially, where the rate equals the negative of the largest Lyapunov exponent of the unperturbed system. We compare the response to optimal forcing with random forcing and find that the optimal forcing is particularly effective if the largest Lyapunov exponent is significantly larger than the other Lyapunov exponents. However, if the largest Lyapunov exponent is much larger than unity, then the optimal forcing decreases rapidly and is only as effective as a single-push forcing.

An intermittent nonlinear map generating subdiffusion is investigated. Computer simulations show that the generalized diffusion coefficient of this map has a fractal, discontinuous dependence on control parameters. An amended continuous time random-walk theory well approximates the coarse behavior of this quantity in terms of a continuous function. This theory also reproduces a full suppression of the strength of diffusion, which occurs at the dynamical transition from normal to anomalous diffusion. Similarly, the probability density function of this map exhibits a nontrivial fine structure while its coarse functional form is governed by a time fractional diffusion equation. A more detailed understanding of the irregular structure of the generalized diffusion coefficient is provided by an anomalous Taylor-Green-Kubo formula establishing a relation to de Rham–type fractal functions.

We study the relaxation of a passive scalar towards the uniform equilibrium distribution in an advection-diffusion problem where the phase space for the pure advection problem is a mixture of chaotic domains and elliptic islands. Since the advection-diffusion problem is linear, the relaxation can be characterized by the eigenvalues and eigenmodes of the evolution operator. Almost degenerate eigenvalues then give rise to deviations from simple exponential decay behavior. We show by example that the corresponding eigenmodes can be supported by islands or weakly connected chaotic domains. These theoretical considerations are related to some experimental observations in two-dimensional flows.

Energy landscape approaches have become increasingly popular for analyzing a wide variety of chemical physics phenomena. Basic to many of these applications has been the inherent structure mapping, which divides up the potential energy landscape into basins of attraction surrounding the minima. Here, we probe the nature of this division by introducing a method to compute the basin area distribution and applying it to some archetypal supercooled liquids. We find that this probability distribution is a power law over a large number of decades with the lower-energy minima having larger basins of attraction. Interestingly, the exponent for this power law is approximately the same as that for a high-dimensional Apollonian packing, providing further support for the suggestion that there is a strong analogy between the way the energy landscape is divided into basins, and the way that space is packed in self-similar, space-filling hypersphere packings, such as the Apollonian packing. These results suggest that the basins of attraction provide a fractal-like tiling of the energy landscape, and that a scale-free pattern of connections between the minima is a general property of energy landscapes.

We study the scaling properties of the solid-on-solid front of the infinite cluster in two-dimensional gradient percolation. We show that such an object is self-affine with a Hurst exponent equal to $2∕3$ up to a cutoff length $\sim g^{-4∕7}$, where $g$ is the gradient. Beyond this length scale, the front position has the character of uncorrelated noise. Importantly, the self-affine behavior is robust even after removing local jumps of the front. The previously observed multiaffinity is due to the dominance of overhangs at small distances in the structure function. This is a crossover effect.

We investigate a new screwlike liquid-crystalline ordering in solutions of helical biopolymers and its influence on the state of individual molecules. In the resulting mesophase translational and rotational motions of molecules are coupled in screw fluctuations. We show that in contrast to the case of conventional chiral liquid crystals the elastic distortion does not twist the screw order but leads to overwinding of individual helical molecules. This explains the peculiarities of high-density DNA mesophases.

Shot noise processes displaying the Noah effect are of infinite variance. Hence the measurement of the correlation structure of such processes—indicating whether the Joseph effect is displayed as well—cannot be conducted via their autocovariance functions (which are undefined). To circumvent this problem, a Poisson-based analysis applicable to shot noise processes with infinite variance, as well as to shot noise processes with divergent noise levels, is developed. A Poissonian autocovariance function, which characterizes the process-distribution of general shot noise processes, is introduced. In particular, this function governs and quantifies the processes’ stationary structure (amplitudal behavior) and correlation structure (temporal behavior). The “Poissonian methodology” developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the Noah and Joseph effects.

Simulations are used to address the effects of oscillating shear strain on jammed systems composed of spherical particles. The simulations show that shear oscillations with amplitudes of more than a few percent lead to substantial crystallization of the system. To ensure that the conclusions are independent of the simulation methodology, a range of simulations are carried out that use both molecular dynamics and athermal dynamics methods, soft and hard potentials, potentials with and without attractive forces, and systems with and without surrounding walls. The extent of crystallization is monitored primarily by the $Q_6$ order parameter, but also in some simulations by the potential energy and the radial distribution function, and by direct visual inspection. A mechanism is proposed for shear-induced crystallization of jammed systems, based on fold catastrophes of the free-energy landscape.

We consider the escape of particles located in the middle well of a symmetric triple well potential driven sinusoidally by two forces such that the potential wells rock as in stochastic resonance and the height of the potential barrier oscillates symmetrically about a mean as in resonant activation. It has been shown that depending on their phase difference the application of these two synchronized signals may lead to a splitting of time averaged Kramers’ escape rate and a preferential product distribution in a parallel chemical reaction in the steady state.

We study the influence of birefringence on incoherent polarized light transport in turbid media. In particular, Mueller matrices backscattered by a diffusing medium are modified by the birefringence of the suspending phase. We study this effect both theoretically, through Monte Carlo simulations, and experimentally with a highly birefringent xanthane solution in which particles are added at various concentrations to modify its turbidity. Comparisons between experiments on flow-induced birefringence of the xanthane solution with or without particles are in good agreement and show the capability of measuring birefringence in turbid media through analysis of Mueller matrices.

The scale-free, long-range correlations detected in DNA sequences contrast with characteristic lengths of genomic elements, being particularly incompatible with the isochores (long, homogeneous DNA segments). By computing the local behavior of the scaling exponent $\alpha$ of detrended fluctuation analysis (DFA), we discriminate between sequences with and without true scaling, and we find that no single scaling exists in the human genome. Instead, human chromosomes show a common compositional structure with two characteristic scales, the large one corresponding to the isochores and the other to small and medium scale genomic elements.

Motived by the work of Li and Meneveau [Phys. Rev. Lett. 95, 164502 (2005)], we propose and solve a model for the Lagrangian evolution of both longitudinal and transverse velocity and temperature increments for Boussinesq convection. From this model, the short-time evolution of an initially imposed Gaussian joint probability density function (PDF) of both velocity and temperature increments is computed analytically and the trend to non-Gaussian statistics shown in a quantitative way. Predictions for moments of the joint PDF are obtained and their behavior analyzed with respect to known experimental and numerical results. The obtained results do not depend on the model free parameters, a fact in favor of their robustness.

We report here the results of numerical investigation of noise effects on the propagation in a nonlinear waveguide modeled by a discrete electrical line. Considering a periodic signal of frequency exceeding the natural cutoff frequency of this system, we show that noise can be used to trigger soliton generation in the medium. Besides the classical stochastic resonance signature exhibited by each oscillator of the network, our simulation results reveal in particular that the signal-to-noise ratio remains almost constant in the whole network for an appropriate amount of noise. This interesting feature insures for the generated solitons a quality preserved propagation along the network.

To identify and understand the effect of pressure on turbulence velocity gradient dynamics, we study the velocity gradient evolution with the inviscid three-dimensional Burgers equation and the restricted Euler equation (REE). While the REE represents the incompressible limit of turbulence, the Burgers equation is taken to be the infinite Mach number model for the Navier-Stokes equation wherein the time scale of flow inertia is very small compared to that of pressure. Analytical fixed-point solutions for the velocity gradient tensor are obtained in the two cases. The results are compared and contrasted to isolate the role of pressure in shaping velocity gradient behavior. Of particular interest is the influence of pressure on (i) the strain rate eigenvalues; (ii) the sign of the intermediate principal strain rate; (iii) the tendency of vorticity to align with the intermediate principal strain rate; and (iv) the energy cascade direction. Importantly, the study provides valuable insight into the velocity gradient dynamics in highly compressible turbulence.

In this paper, we study the critical ratio between the amplitudes of two overtaking solitary waves on a layer of water with uniform depth. At the center of encounter, the wave profile is fore-and-aft symmetric, but it could have a single peak or double peaks. The critical ratio separates these two regimes. At the critical point, the wave peak is flat with zero slope and curvature. We solve the full water wave problem numerically by using a fully nonlinear and highly dispersive Boussinesq model. The model is numerically justified to be a good approximation of the Euler equations for solitary waves with very large amplitude. For small amplitude water waves, our calculated critical ratio reduces to the well-known result of 3 predicted by the Korteweg–de Vries equation, a weakly nonlinear and weakly dispersive model. For large amplitude water waves, the nonlinear effect is significant; we find that the critical ratio deviates significantly from 3. For water waves with very high amplitude, e.g., 0.6 relative wave height, the critical ratio could be as large as 4. Our result suggests that higher-order nonlinear and dispersive effects are important when modeling the strong interaction between large amplitude water waves.

The compression of femtosecond laser pulses by linear quasiperiodic and periodic photonic multilayer structures is studied both experimentally and theoretically. We compare the compression performance of a Fibonacci and a periodic structure with similar total thickness and the same number of layers, and find the performance to be higher in the Fibonacci case, as predicted by numerical simulation. This compression enhancement takes place due to the larger group velocity dispersion at a defect resonance of the transmission spectrum of the Fibonacci structure. We demonstrate that the Fibonacci structure with the thickness of only $2.8\mu \mathrm{m}$ can compress a phase-modulated laser pulse by up to 30%. The possibility for compression of laser pulses with different characteristics in a single multilayer is explored. The operation of the compressor in the reflection regime has been modeled, and we show numerically that the reflected laser pulse is subjected to real compression: not only does its duration decrease but also its amplitude rises.

The available numerical algorithms for trend removal require a direct subjective intervention in choosing critical parameters. In this paper an algorithm is presented, which needs no initial subjective assumptions. Monotone trends are approximated by piecewise linear curves obtained by dividing into subintervals the signal values interval, not the time interval. The slope of each linear segment of the estimated trend is proportional to the average one-step displacement of the signal values included into the corresponding subinterval. The evaluation of the trend removal is performed on statistical ensembles of artificial time series with the random component given by realizations of autoregressive of order one stochastic processes or by fractional Brownian motions. The accuracy of the algorithm is compared with that of two well-tested methods: polynomial fitting and a nonparametric method based on moving average. For stationary noise the results of the algorithm are slightly better, but for nonstationary noise the preliminary results indicate that the polynomial fitting has the best accuracy. As a verification on a real time series, the time periods with monotone variation of global average temperature over the last $1800\text{years}$ are established. The removal of a nonmonotone trend is also briefly discussed.