Two trains of light pulses at periods that are equally shifted from the harmonics of a missing fundamental are combined in a nonlinear crystal. As a result of a noncollinear phase-matched second-order nonlinear generation, a new train of pulses is obtained. When the temporal width of the input pulses is large, the frequency of the resulting pulse train follows the observations from classical experiments on the perception of virtual pitch by the brain. On the other hand, when the width of the input pulses is small, the generated pulse train exhibits much lower frequencies, analogous to those observed in the motor neural system. Our experimental setup allows us to explore, systematically and continuously, the transition between these two regimes, while at the same time to demonstrate that the functionalities that have been observed in the nervous system are similar to the ones we observe from coincidence detection in quadratic nonlinear systems.

We study the mean escape time in a market model with stochastic volatility. The process followed by the volatility is the Cox, Ingersoll, and Ross process which is widely used to model stock price fluctuations. The market model can be considered as a generalization of the Heston model, where the geometric Brownian motion is replaced by a random walk in the presence of a cubic nonlinearity. We investigate the statistical properties of the escape time of the returns, from a given interval, as a function of the three parameters of the model. We find that the noise can have a stabilizing effect on the system, as long as the global noise is not too high with respect to the effective potential barrier experienced by a fictitious Brownian particle. We compare the probability density function of the return escape times of the model with those obtained from real market data. We find that they fit very well.

We studied the effects of spatial inhomogeneities on inwardly rotating spiral waves in a typical type of oscillatory medium using the complex-Ginzburg-Landau equation. With a small degree of the inhomogeneity in the medium, the slower inward spiral always suppressed a faster spiral; when the inhomogeneity exceeded a critical value, however, a transition occurred to the coexistence of multiple inward spirals, insulated by regions of highly disordered wave break. The occurrence of this transition is examined theoretically and shown to be due to the Eckhaus instability.

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarized intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarized intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarized intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.

We investigate the ground state of a system of interacting particles in small nonlinear lattices with $M⩾3$ sites, using as a prototypical example the discrete nonlinear Schrödinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground-state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground-state (modulational) instability appears to be intimately connected with a nonstandard (“double transcritical”) type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.

We investigate the nonlinear dynamics in a trimer, described by the one-dimensional discrete nonlinear Schrödinger equation, with periodic boundary conditions in the presence of a single on-site defect. We make use of numerical continuation to study different families of stationary and periodic solutions in this Hamiltonian system. Taking into account a suitable Poincaré section, we are able to study the dynamics of a generic thin stochastic layer in this conservative system. Our results strongly suggest that intermittency, on the one hand, and transport between two almost invariant sets, on the other hand, are relevant features of the chaotic dynamics. This behavior arises as a result of the formation of the above mentioned stochastic layer connecting two hyperbolic fixed points of the Poincaré return map. We find that the transit times, the time intervals to traverse some suitable sets in phase space, generate $1∕f$ noise. Its origin is explained in terms of a hopping mechanism in a suitable discrete state space of transit times. A qualitatively similar behavior is also found in the standard map, which shows the generic nature of this mechanism. As a physical application of our results, we consider a possible experiment in a ring of weakly coupled Bose-Einstein condensates (BECs) with attractive interactions, where intermittent bursts of the relative phase of two spatially symmetric BECs take place.

The Rayleigh-Taylor instability of a magnetic fluid superimposed on a nonmagnetic liquid of lower density may be suppressed with the help of a spatially homogeneous magnetic field rotating in the plane of the undisturbed interface. Starting from the complete set of Navier-Stokes equations for both liquids a Floquet analysis is performed which consistently takes into account the viscosities of the fluids. Using experimentally relevant values of the parameters we suggest to use this stabilization mechanism to provide controlled initial conditions for an experimental investigation of the Rayleigh-Taylor instability.

A method to obtain the probability distribution of the interarrival times of jump occurrences in systems driven by state-dependent Poisson noise is proposed. Such a method uses the survivor function obtained by a modified version of the master equation associated to the stochastic process under analysis. A model for the timing of human activities shows the capability of state-dependent Poisson noise to generate power-law distributions. The application of the method to a model for neuron dynamics and to a hydrological model accounting for land-atmosphere interaction elucidates the origin of characteristic recurrence intervals and possible persistence in state-dependent Poisson models.

In our studies, we focus on the reaction of cholesteric side-chain liquid single-crystal elastomers (SCLSCEs) to static external mechanical and electric fields. By means of linearized continuum theory, different geometries are investigated: The mechanical forces are oriented in a direction either parallel or perpendicular to the axis of the cholesteric helix such that they lead to a compression or dilation of the elastomer. Whereas only a homogeneous deformation of the system is found for the parallel case, perpendicularly applied mechanical forces cause either twisting or untwisting of the cholesteric helix. This predominantly depends on the direction in which the director of the cholesteric phase is anchored at the boundaries of the elastomer, and on the sign of a material parameter that describes how deformations of the elastomer couple to the relative rotations between the elastomer and the director. It is also this material parameter that leads to an anisotropy of the mechanical reaction of the system to compression and dilation, due to the liquid crystalline order. The effect of an external electric field is studied when applied parallel to the helix axis of a perfect electric insulator. Here an instability arises at a threshold value of the field amplitude, where the latter results from a competition between the effects of the external electric field on the one hand and the influences of the boundaries of the system, the cholesteric order, and the coupling between the director and the polymer network on the other hand. The instability is either homogeneous in space in the directions perpendicular to the external electric field and includes homogeneous shearing, or, for certain values of the material parameters, there arise undulations of the elastomer and the director orientation perpendicular to the direction of the external electric field at onset. This describes a qualitatively new phenomenon not observed in cholesteric systems yet, as these undulations are due to the coupling of the director orientation to the deformations of the elastomer. Furthermore, we find that theoretically it should be possible to synthesize cholesteric and nematic SCLSCEs in which the director can be reoriented by an external electric field without macroscopically distorting the sample; the latter applies for one specific set of values for the material parameters involved. We also propose ways of experimental access to the material parameters in the general situation.

We describe multifrequency phase synchronization in epileptic seizures. Using magnetoencephalographic recordings from three patients suffering generalized seizures, the evidence is presented that, in addition to the commonly studied 1:1 frequency locking, there exists complex multifrequency coordination that, in some cases, follows a classical “devil’s staircase.” Within the limitations of observing this phenomenon in a clinical experimental setting, these observations reveal that in pathological brain activity, complex frequency locking can be found similar to that identified in certain pathological cardiac re-entrant arrhythmias. This may suggest the existence of similar re-entrant mechanisms active in cerebral neocortex during epileptic seizures.

We analyze the statistical distribution function for the height fluctuations of brittle fracture surfaces using extensive experimental data sampled on widely different materials and geometries. We compare a direct measurement of the distribution to an analysis based on the structure functions. For length scales $\delta$ larger than a characteristic scale $\Lambda$ that corresponds to a material heterogeneity size, we find that the distribution of the height increments $\mathrm{\Delta }h=h(x+\delta )-h\left(x\right)$ is Gaussian and monoaffine, i.e., the scaling of the standard deviation $\sigma$ is proportional to ${\delta }^{\zeta }$ with a unique roughness exponent. Below the scale $\Lambda$ we observe a deviation from a Gaussian distribution and a monoaffine behavior. We discuss for the latter, the relevance of a multiaffine analysis and the influences of the discreteness resulting from material microstructures or experimental sampling.

In order to investigate the origin of large price fluctuations, we analyze stock price changes of ten frequently traded NASDAQ stocks in the year 2002. Though the influence of the trading frequency on the aggregate return in a certain time interval is important, it cannot alone explain the heavy-tailed distribution of stock price changes. For this reason, we analyze intervals with a fixed number of trades in order to eliminate the influence of the trading frequency and investigate the relevance of other factors for the aggregate return. We show that in tick time the price follows a discrete diffusion process with a variable step width while the difference between the number of steps in positive and negative direction in an interval is Gaussian distributed. The step width is given by the return due to a single trade and is long-term correlated in tick time. Hence, its mean value can well characterize an interval of many trades and turns out to be an important determinant for large aggregate returns. We also present a statistical model reproducing the cumulative distribution of aggregate returns. For an accurate agreement with the empirical distribution, we also take into account asymmetries of the step widths in different directions together with cross correlations between these asymmetries and the mean step width as well as the signs of the steps.

We study the behaviour of steady-state solutions of a two-component flame filament system subject to chaotic mixing. This system exhibits a saddle-node bifurcation at a critical Damköhler number. We analyze the system through a one-dimensional phenomenological lamellar model. We present a nonperturbative technique, which allows us to describe the behaviour of the reduced lamellar model near the saddle node bifurcation. The influence of the Lewis number on the solution behaviour is investigated. We present a simple empirical formula for the wave speed valid for large Damköhler and large Lewis numbers. This formula allows us to describe the solution far away from the bifurcation. Numerical simulations show good agreement with the results.

We discuss synchronization thresholds in an array of nondiagonally coupled oscillators. We argue that nondiagonal coupling can cause the appearance or disappearance of desynchronous windows in the coupling parameter space. Such a phenomenon is independent of the motion character (periodic or chaotic) of the isolated node system. A mechanism governing this phenomenon is explained and its influence on the global network dynamics is analyzed.

We present the results of an experimental study on the transition to spiral vortices in flow between concentric counter-rotating cylinders in the presence of an axial through-flow, i.e., in spiral Poiseuille flow. The experiments were performed in an apparatus having an aspect ratio $\Gamma =L∕d=22.8$ ($L$ axial length, $d$ gap width between cylinders) and end plates enabling an in and outflow of mass. As a result of an applied axial through-flow the “classical” Hopf bifurcation to spiral vortices (SPI) splits up and a primary and secondary branch of down and upstream propagating SPI, respectively, as well as a transient quasiperiodic flow appear. Downstream propagating SPI resulting from the primary supercritical Hopf bifurcation are either convectively or absolutely unstable. The bifurcation structure observed in this open flow experiment is in qualitative agreement with predictions from theory of Hopf bifurcation with broken reflection symmetry [J. D. Crawford and E. Knobloch, Nonlinearity 1, 617 (1988)] and also in quantitative agreement with results from recent numerical calculations [A. Pinter, M. Lücke, and C. Hoffmann, Phys. Rev. E 67, 026318 (2003); C. Hoffmann, M. Lücke, and A. Pinter, Phys. Rev. E 69, 056309 (2004)].

The energy transfer and small-scale intermittency in decaying turbulence in four dimensions (4D) are studied by direct numerical simulation and by spectral theory in comparison with three dimensions (3D). The energy transfer is more efficient in 4D than in 3D, hence the exponent of energy decay is larger. The Kolmogorov constant is 1.31, which is smaller than 1.72 in 3D. The longitudinal third-order structure function is confirmed to be governed by a $1∕2$ law, $-(1∕2)\overline{\epsilon }r$, instead of a $4∕5$ law in 3D. The intermittency is weaker in 4D for the total dissipation rate $\nu {\sum }_{i,j}{(\partial u_j∕\partial x_i)}^2$ and the associated velocity difference (spherical velocity difference) on scale $r$ than in 3D, while it is slightly stronger for the surrogated dissipation rate $\nu {(\partial u_1∕\partial x_1)}^2$ and the associated longitudinal velocity difference. The scaling exponents of the spherical and longitudinal velocity differences are also evaluated, indicating that the spherical velocity difference is less intermittent in 4D, while the longitudinal difference is more intermittent in 4D. The distribution of the eigenvalues of the strain tensor is also examined. It was also found that the normalized variance of the pressure gradient ${\overline{\epsilon }}^{-3∕2}{\nu }^{-1∕2}⟨{(\nabla p)}^2⟩$ in 4D is smaller than in 3D. The roles of the incompressibility condition, the pressure gradient, and the intermittency in $d$-dimensional turbulence are examined, and the importance of the longitudinal component of turbulent velocity field in the energy transfer toward small scales are discussed. Burgers turbulence as the asymptote of turbulence in large dimension is suggested.

We characterize fully the polarization dynamics of the fundamental vector soliton of isotropic Kerr materials by measuring the Stokes parameters of an elliptically polarized self-trapped optical beam propagating in a slab planar waveguide. Our experiment clearly shows that this two-component spatial vector soliton exhibits both the so-called ellipse rotation and curved-shape ellipticity that are due to the self-induced nonlinear birefringence between the two components of the vector soliton. The polarization of the vector soliton is accurately determined both in the transverse and longitudinal directions and comparisons with numerical simulations based on two coupled nonlinear Schrödinger equations provide an excellent quantitative agreement. Spatiotemporal numerical simulations that take into account the finite pulse duration of the experimental input optical beam must, however, be used to match rigorously the measured state of polarization of the vector soliton.

We propose an equivalence class of nonstationary Gaussian stochastic processes defined in the wavelet domain. These processes are characterized by means of wavelet multipliers and exhibit well-defined time-dependent spectral properties. They allow one to generate realizations of any wavelet spectrum. Based on this framework, we study the estimation of continuous wavelet spectra, i.e., we calculate variance and bias of arbitrary estimated continuous wavelet spectra. Finally, we develop an areawise significance test for continuous wavelet spectra to overcome the difficulties of multiple testing; it uses basic properties of continuous wavelet transform to decide whether a pointwise significant result is a real feature of the process or indistinguishable from typical stochastic fluctuations. This test is compared to the conventional one in terms of sensitivity and specificity. A software package for continuous wavelet spectral analysis and synthesis is presented.

The influence of the crystal field on the phase diagrams of the bilayer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants $J_1$ and $J_2$ of the two layers and interlayer coupling constant $J_3$ between the layers for given values of the coordination number $q$ by using the recursion relation scheme. The six distinct ground-state configurations of the model are obtained on the $(J_2∕J_1,J_3∕q{J}_1)$ plane with $J_1>0$, the ferromagnetic coupling, for given values of the crystal field. Then, the phase diagram of the system is obtained on the $(kT∕J_1,J_3∕J_1)$ plane for given values of the crystal field and $\alpha =J_2∕J_1$ with $q=4$ corresponding to the square lattice in real lattice systems. It was found that the system presents both first- and second-order phase transitions, therefore, tricritical points. The paramagnetic phase was also divided into two phases, $P_{+}$ and $P_{-}$, by studying the thermal behavior of the quadrupolar moments of the two layers.