We have studied the effect of random long-range connections in chaotic thermosensitive neuron networks with each neuron being capable of exhibiting diverse bursting behaviors, and found stochastic synchronization and optimal spatiotemporal patterns. For a given coupling strength, the chaotic burst-firings of the neurons become more and more synchronized as the number of random connections (or randomness) is increased and, rather, the most pronounced spatiotemporal pattern appears for an optimal randomness. As the coupling strength is increased, the optimal randomness shifts towards a smaller strength. This result shows that random long-range connections can tame the chaos in the neural networks and make the neurons more effectively reach synchronization. Since the model studied can be used to account for hypothalamic neurons of dogfish, catfish, etc., this result may reflect the significant role of random connections in transferring biological information.

We introduce a model for information spreading among a population of $N$ agents diffusing on a square $L\times L$ lattice, starting from an informed agent (Source). Information passing from informed to unaware agents occurs whenever the relative distance is $⩽1$. Numerical simulations show that the time required for the information to reach all agents scales as $N^{-\alpha }{L}^{\beta }$, where $\alpha$ and $\beta$ are noninteger. A decay factor $z$ takes into account the degeneration of information as it passes from one agent to another; the final average degree of information of the population ${\mathcal{I}}_{av}\left(z\right)$ is thus history dependent. We find that the behavior of ${\mathcal{I}}_{av}\left(z\right)$ is nonmonotonic with respect to $N$ and $L$ and displays a set of minima. Part of the results are recovered with analytical approximations.

In this work we present a model for the propagation of culture on networks of different topology and by considering different underlying dynamics. We extend a previous model proposed by Axelrod by letting a majority govern the dynamics of changes. This in turn allows us to define a Lyapunov functional for the system.

When two characteristically different chaotic oscillators are coupled, generalized synchronization can occur. Motivated by the phenomena that common noise can induce and enhance complete synchronization or phase synchronization in chaotic systems, we investigate the effect of noise on generalized chaotic synchronization. We develop a phase-space analysis, which suggests that the effect can be system dependent in that common noise can either induce/enhance or destroy generalized synchronization. A prototype model consisting of a Lorenz oscillator coupled with a dynamo system is used to illustrate these phenomena.

We study a one-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DB's) can be explicitly constructed by an exact separation of their time and space dependence. Introducing parametric periodic driving, we first show how a variety of such DB's can be obtained by selecting spatial profiles from the homoclinic orbits of an invertible map and combining them with initial conditions chosen from the Poincaré surface of section of a simple Duffing's equation. Placing then our initial conditions at the center of the islands of a major resonance, we demonstrate how the corresponding DB can be stabilized by varying the amplitude of the driving. We thus discover around elliptic points a large region of quasiperiodic breathers, which are stable for very long times. Starting with initial conditions close to the elliptic point at the origin, we find that as we approach the main chaotic layer, a quasiperiodic breather either destabilizes by delocalization or turns into a chaotic breather, with an evidently broadbanded Fourier spectrum before it collapses. For some breather profiles stable quasiperiodic breathers exist all the way to the separatrix of the Duffing equation, indicating the presence of large regions of tori around the DB solution in the multidimensional phase space. We argue that these strong localization phenomena are due to the absence of phonon resonances, as there are no linear dispersion terms in our lattices. We also show, however, that these phenomena persist in more realistic physical models, in which weak linear dispersion is included in the equations of motion, with a sufficiently small coefficient.

Feedback control of a subcritical oscillatory instability is investigated in the framework of a globally-controlled complex Ginzburg-Landau equation that describes the nonlinear dynamics near the instability threshold. The control is based on a feedback loop between the system linear growth rate and the maximum of the amplitude of the emerging pattern. It is shown that such control can suppress the blow up and result in the formation of spatially localized pulses similar to oscillons. In the one-dimensional case, depending on the values of the linear and nonlinear dispersion coefficients, several types of the pulse dynamics are possible in which the computational domain contains: (i) a single stationary pulse; (ii) several coexisting stationary pulses; (iii) competing pulses that appear one after another at random locations so that at each moment of time there is only one pulse in the domain; (iv) spatiotemporally chaotic system of short pulses; (v) spatially-synchronized pulses. Similar dynamic behavior is found also in the two-dimensional case. The effect of the feedback delay is also studied. It is shown that the increase of the delay leads to an oscillatory instability of the pulses and the formation of pulses with oscillating amplitude.

The diffusion-limited trapping reaction kinetics of the growth of the depletion zone within and around a “slit-shaped” trap in a flat microchannel was studied experimentally and numerically. In the experiment, an ellipse-shaped laser beam acted as a slit trap in a long, flat capillary, and the trapping reaction is photobleaching of fluorescein dye. The parameter studied was the $\theta$ distance, i.e., the distance from the trap to the point where the reactant concentration has been locally depleted to the specific survival fraction $\left[\theta \right]$ of its initial bulk value. When the trap is perfect, then, due to the geometry of the trap and the reactor, as many as three time regimes can be found, with up to two crossover transitions. The number of crossovers is determined by the relative sizes of the trap and the microreactor. In the case of two crossovers, we show that the first crossover relates to the length of the trap, while the second crossover relates to the width of the reactor. When the slit trap is imperfect and its width cannot be neglected, as is the case in the experiments, a nontrivial early behavior is observed, followed by two regions in time, separated by a single crossover only.

The problem of potential flow inside an evaporating liquid line, shaped as an infinitely long cylindrical segment lying on a flat surface, is considered and an analytical solution is obtained for any contact angle in $(0,\pi )$. In this way, microflow details inside linear liquid bodies evaporating on hydrophilic, hydrophobic, and strongly hydrophobic substrates can now be obtained. The mathematical formulation employs the velocity potential and stream function formulations in bipolar coordinates and the solution is obtained using the technique of Fourier transform. Both pinned and depinned contact lines are considered. The solution is applicable to any evaporation mechanism but for illustration purposes numerical results are presented here for the particular case of kinetically controlled evaporation. For hydrophilic substrates, the flow inside the evaporating liquid line is directed towards the edges for pinned contact lines, thus, promoting a coffee stain effect. The opposite flow direction is observed for depinned contact lines. However, for strongly hydrophobic substrates, flow is directed outwards for both pinned and depinned contact lines, but owing to its low magnitude compared to that on hydrophilic substrates, a craterlike colloidal deposit should be expected rather than a ringlike deposit, in agreement with experimental observations.

We study the pair correlation function $g\left(r\right)$ for zero-temperature, disordered, soft-sphere packings just above the onset of jamming. We find distinct signatures of the transition in both the first and split second peaks of this function. As the transition is approached from the jammed side (at higher packing fraction) the first peak diverges and narrows on the small-$r$ side to a $\delta$ function. On the high-$r$ side of this peak, $g\left(r\right)$ decays as a power law. In the split second peak, the two subpeaks are both singular at the transition, with power-law behavior on their low-$r$ sides and step-function drop-offs on their high-$r$ sides. These singularities at the transition are reminiscent of empirical criteria that have previously been used to distinguish glassy structures from liquid ones.

We present an experimental study of the mechanical impulse propagation through a horizontal alignment of elastic spheres of progressively decreasing diameter ${\varphi }_n$: namely, a tapered chain. Experimentally, the diameters of spheres which interact via the Hertz potential are selected to keep as close as possible to an exponential decrease, ${\varphi }_{n+1}=(1-q){\varphi }_n$, where the experimental tapering factor is either $q_1\simeq 5.60\%$ or $q_2\simeq 8.27\%$. In agreement with recent numerical results, an impulse initiated in a monodisperse chain (a chain of identical beads) propagates without shape changes and progressively transfers its energy and momentum to a propagating tail when it further travels in a tapered chain. As a result, the front pulse of this wave decreases in amplitude and accelerates. Both effects are satisfactorily described by the hard-sphere approximation, and basically, the shock mitigation is due to partial transmissions, from one bead to the next, of momentum and energy of the front pulse. In addition when small dissipation is included, better agreement with experiments is found. A close analysis of the loading part of the experimental pulses demonstrates that the front wave adopts a self-similar solution as it propagates in the tapered chain. Finally, our results corroborate the capability of these chains to thermalize propagating impulses and thereby act as shock absorbing devices.

Experiments reported by Blake et al. [Phys. Fluids., 11, 1995 (1999)] suggest that the dynamic contact angle formed between the free surface of a liquid and a moving solid boundary at a fixed contact-line speed depends on the flow field and geometry near the moving contact line. We examine quantitatively whether or not it is possible to attribute this effect to the bending of the free surface due to hydrodynamic stresses acting upon it and hence interpret the results in terms of the so-called “apparent” contact angle. It is shown that this is not the case. Numerical analysis of the problem demonstrates that, at the spatial resolution reported in the experiments, the variations of the “apparent” contact angle (defined in two different ways) caused by variations in the flow field, at a fixed contact-line speed, are too small to account for the observed effect. The results clearly indicate that the actual (macroscopic) dynamic contact angle—i.e., the one used in fluid mechanics as a boundary condition for the equation determining the free surface shape—must be regarded as dependent not only on the contact-line speed but also on the flow field and geometry in the vicinity of the moving contact line.

We consider as a mathematical model of human cortical electrical activity a system of fourteen ordinary differential equations. With appropriate parameters, the model produces activity characteristic of a seizure. To prevent such seizures, we incorporate feedback controllers into the model dynamics. We show that three controllers—a linear feedback controller, a differential controller, and a filter controller—can be used to eliminate seizing activity in the model system. We show how bifurcations induced by the linear controller alter those present in the original dynamics.

We investigate the stimulus-dependent tuning properties of a noisy ionic conductance model for intrinsic subthreshold oscillations in membrane potential and associated spike generation. Upon depolarization by an applied current, the model exhibits subthreshold oscillatory activity with an occasional spike generation when oscillations reach the spike threshold. We consider how the amount of applied current, the noise intensity, variation of maximum conductance values, and scaling to different temperature ranges alter the responses of the model with respect to voltage traces, interspike intervals and their statistics, and the mean spike frequency curves. We demonstrate that subthreshold oscillatory neurons in the presence of noise can sensitively and also selectively be tuned by the stimulus-dependent variation of model parameters.

We report successful manipulation of the noise provoked spiking behavior using delayed feedback control. Experiments were performed in a three electrode electrochemical cell under potentiostatic conditions. The uncontrolled system exhibited noise invoked oscillations whose regularity was quantified using normalized variance (NV). Superimposing delayed feedback, for appropriate values of delay $\left(\tau \right)$, an enhancement in the regularity of the spike sequence was attained. Numerical simulations corroborated experimental observations.

We report results of investigations of a high-speed drainage of thin aqueous films squeezed between randomly nanorough surfaces. A significant decrease in the hydrodynamic resistance force as compared with that predicted by Taylor’s equation is observed. However, this reduction in force does not represent the slippage. The measured force is exactly the same as that between equivalent smooth surfaces obeying no-slip boundary conditions, but located at the intermediate position between peaks and valleys of asperities. The shift in hydrodynamic thickness is shown to be independent of the separation and/or shear rate. Our results disagree with previous literature data reporting very large and shear-dependent boundary slip for similar systems.

The space-time epidemic-type aftershock sequence model is a stochastic branching process in which earthquake activity is classified into background and clustering components and each earthquake triggers other earthquakes independently according to certain rules. This paper gives the probability distributions associated with the largest event in a cluster and their properties for all three cases when the process is subcritical, critical, and supercritical. One of the direct uses of these probability distributions is to evaluate the probability of an earthquake to be a foreshock, and magnitude distributions of foreshocks and nonforeshock earthquakes. To verify these theoretical results, the Japan Meteorological Agency earthquake catalog is analyzed. The proportion of events that have 1 or more larger descendants in total events is found to be as high as about 15%. When the differences between background events and triggered event in the behavior of triggering children are considered, a background event has a probability about 8% to be a foreshock. This probability decreases when the magnitude of the background event increases. These results, obtained from a complicated clustering model, where the characteristics of background events and triggered events are different, are consistent with the results obtained in [Ogata et al., Geophys. J. Int. 127, 17 (1996)] by using the conventional single-linked cluster declustering method.

Exact finite-size scaling functions of the interfacial tensions are obtained for the Ising model with isotropic coupling on a set of $M\times N$ planar lattices, including square (sq), plane triangular (pt), and honeycomb (hc) lattices. The analyses of transitive behaviors at criticality revise the knowledge of the interfacial tensions as a function of the aspect ratio defined by $R=M∕N$ for $R$ approaching to zero gradually. The amplitudes of the interfacial tensions for the sq, pt, and hc lattices are further shown to have relative proportions $1:\sqrt{3}∕2:\sqrt{3}$ which are related to the aspect ratios for the three lattices to have similar domains.

We consider two different approaches, to which we refer to as renewal and modulation, to generate time series with a nonexponential distribution of waiting times. We show that different time series with the same waiting time distribution are not necessarily statistically equivalent, and might generate different physical properties. Renewal generates aging and anomalous scaling, while modulation yields no significant aging and either ordinary or anomalous diffusion, according to the dynamic prescription adopted. We show, in fact, that the physical realization of modulation generates two classes of events. The events of the first class are determined by the persistent use of the same exponential time scale for an extended lapse of time, and consequently are numerous; the events of the second class are identified with the abrupt changes from one to another exponential prescription, and consequently are rare. The events of the second class, although rare, determine the scaling of the diffusion process, and for this reason we term them as crucial events. According to the prescription adopted to produce modulation, the distribution density of the time distances between two consecutive crucial events might have, or not, a diverging second moment. In the former case the resulting diffusion process, although going through a transition regime very extended in time, will eventually become anomalous. In conclusion, modulation rather than ruling out the action of renewal events, produces crucial events hidden by clouds of exponential events, thereby setting the challenge for their identification.

The disagreement of the scaling of the correlation length $\xi$ between experiment and the Ginzburg-Landau (GL) model for domain chaos was resolved. The Swift-Hohenberg (SH) domain chaos model was integrated numerically to acquire test images to study the effect of a finite image size on the extraction of $\xi$ from the structure factor (SF). The finite image size had a significant effect on the SF determined with the Fourier-transform (FT) method. The maximum entropy method (MEM) was able to overcome this finite image-size problem and produced fairly accurate SFs for the relatively small image sizes provided by experiments. Correlation lengths often have been determined from the second moment of the SF of chaotic patterns because the functional form of the SF is not known. Integration of several test functions provided analytic results indicating that this may not be a reliable method of extracting $\xi$. For both a Gaussian and a squared SH form, the correlation length $\overline{\xi }\equiv 1∕\sigma$, determined from the variance ${\sigma }^2$ of the SF, has the same dependence on the control parameter $\epsilon$ as the length $\xi$ contained explicitly in the functional forms. However, for the SH and the Lorentzian forms we find $\overline{\xi }\sim {\xi }^{1∕2}$. Results for $\xi$ determined from new experimental data by fitting the functional forms directly to the experimental SF yielded $\xi \sim {\epsilon }^{-\nu }$ with $\nu \simeq \frac{1}{4}$ for all four functions in the case of the FT method, but $\nu \simeq \frac{1}{2}$, in agreement with the GL prediction, in the case of the MEM. Over a wide range of $\epsilon$ and wave number $k$, the experimental SFs collapsed onto a unique curve when appropriately scaled by $\xi$.

We investigate experimentally induction mechanisms in a screw flow of gallium in a toroidal channel. The flow is nonstationary and operated in a spin-down regime: the channel (and fluid) are initially set into solid body rotation; as the channel is stopped the fluid is set into strong helical motion by diverters located inside the channel. In this study, we put a particular emphasis on the induction generated by these helical motions, which are expected to develop over the entire range of turbulent scales. We apply an external magnetic field either perpendicular to the channel axis parallel to it. At large scales the nonlinear induction mechanisms are associated with the Parker stretch and twist effect and with the expulsion due to overall rotation. Induction mechanisms can also originate in the small scale helicity as in the alpha induction effect of mean-field magnetohydrodynamics. Our measurements yield an upper bound for the alpha coefficient, significantly lower than estimates based on dimensional analysis. We discuss the consequences of our observations for the engineering of homogeneous dynamos in the laboratory.