We propose a definition of finite-space Lyapunov exponent. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by showing that, for large classes of chaotic maps, the corresponding finite-space Lyapunov exponent approaches the Lyapunov exponent of a chaotic map when $M\to \infty$, where $M$ is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has pseudochaos if its finite-space Lyapunov exponent tends to a positive number (or to $+\infty$), when $M\to \infty$.

We study the temporal evolution of speckle patterns in transmission of short wave pulses through a disordered waveguide. In the diffuse regime, the short-range spatial structure of speckles is the same as for the continuous-wave (cw) illumination, whereas the long-range correlation between distant speckle spots grows linearly with time and can exceed its cw value. We discuss the physical origin of this phenomenon, compare our results to recent microwave experiments, and suggest that a similar linear growth with time should also be characteristic for other mesoscopic interference phenomena.

New nonlinear evolution equations are derived that generalize those presented in a Letter by Matsuno [Phys. Rev. Lett. 69, 609 (1992)] and a terrain-following Boussinesq system recently deduced by Nachbin [SIAM J Appl. Math. 63, 905 (2003)]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. A Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. The formulation is over a periodically extended domain so that, as an application, it produces efficient Fourier (fast-Fourier-transform algorithm) solvers.

Water permeation through nanometric channels, generally a coupled many-body process, is described in this study by a single collective coordinate. A collective diffusion model is proposed in which water movement at equilibrium is characterized as an unbiased diffusion along this coordinate and water transport in the presence of a chemical potential difference is described as one-dimensional diffusion in a linear potential. The model allows one to determine the osmotic permeability of a water channel from equilibrium simulations.

We present the first experimental observation of modulation instability and spontaneous pattern formation with incoherent white light emitted from an incandescent light bulb. We show experimentally that modulation instability of white light propagating in a noninstantaneous self-focusing medium is a collective effect, where the entire temporal spectrum of the light beam becomes unstable at the same threshold value and collectively forms a pattern with a single periodicity. We experimentally demonstrate that the temporal spectrum of the evolving perturbation self-adjusts to match the collective pattern formation phenomenon.

We analyze a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling motivated by the physics of arrays of nanoscale oscillators. We study the model for the mean field case of all-to-all coupling, deriving results for the onset of synchronization as the coupling or nonlinearity increase, and the fully locked state when all the oscillators evolve with the same frequency.

We model the radial segregation patterns that form in a thin rotating drum partially filled with beads of two sizes. We predict that the waviness (or amplitude-to-wavelength ratio, denoted $w$) of a pattern should be subjected to low-pass filtering with a cutoff waviness $w_c$ that depends strongly on the fill level of the drum. Then we perform experiments and find that $w\le w_c$ for all patterns, in accord with our prediction. We also find that $w=w_c$ for (and only for) steady patterns, and conclude that the waviness of a steady pattern is selected by the low-pass filtering.

A classical simulation of equilibrium thermal emissivity from dispersive, lossy photonic crystals is presented. Normal emission results consistent with those assuming Kirchoff's law are obtained; i.e., a photonic crystal does not emit more than what a blackbody does. Significant enhancement, however, can be achieved over the radiation intensity from a uniform slab, indicating the potential usefulness of photonic crystals in incandescent lighting and thermal photovoltaic applications.

The dynamics of a passive scalar such as a dye in the far dissipative range of fluid turbulence is a central problem in nonlinear physics. An important prediction for this problem was made by Batchelor over 40 years ago and is known as Batchelor's scaling law. We here present strong evidence in favor of this law for the thickness fluctuations in the flow of a soap film past a flat plate. The results also capture the dissipative range of the scalar which turns out to have universal features. The probability density function of the scalar increments and their structure functions come out in nice agreement with theoretical predictions.

We present an experimental study of the coiling instability of a liquid “rope” falling on a solid surface. Coiling can occur in three different regimes—viscous, gravitational, or inertial—depending on the fluid viscosity and density, the fall height, and the flow rate. The competition among the different forces causes the coiling frequency first to decrease and subsequently to increase with increasing height. We also observe an oscillation between two coiling states in the gravitational-to-inertial transitional range, reflecting the multivaluedness of the dependence of coiling frequency on fall height. The data can be rescaled in a universal way, and agree very well with numerically predicted coiling frequencies.

When a rotating rod is brought into a polymer melt or concentrated polymer solution, the meniscus climbs the rod. This spectacular rod climbing is due to the normal stresses present in the polymer fluid and is thus a purely non-Newtonian effect. A similar rod climbing of an interface between two fluids has therefore been taken as a signature that one of the fluids exhibits normal stress effects. We show here, however, that the effect can occur with simple Newtonian fluids: it occurs when a Taylor-Couette instability happens in the less viscous of the two liquids but not in the more viscous one.

We demonstrate experimentally and theoretically that a coherent image of a pure phase object [implemented by a microelectromechanical system (MEMS) micromirror array] may be obtained by use of a spatially incoherent illumination beam. This is accomplished by employing a two-beam source of entangled photons generated by spontaneous parametric down-conversion. One of the beams probes the phase object while the other is scanned. Though each of the beams is, in and of itself, spatially incoherent, the pair of beams exhibits higher-order interbeam coherence.

We report the first direct experimental observation of the fast dynamics (nanosecond scale) of complex two-dimensional transverse patterns in broad area lasers. The laser emission bright peaks forming the transverse patterns are observed to be aperiodically flashing in time with different growing rates. These optical filaments do not move along the cross section during their lifetime, which is close to 2 ns. The experimental observations have also been reproduced by numerical integration of the Maxwell-Bloch equations.

The dependence of the shift of an optical bistability hysteresis curve on the nonlinear phase shift induced by a controlling light is observed in a four-level atomic system of ${}^{87}\mathrm{R}\mathrm{b}$ inside an optical ring cavity. In the process the intensity of the coupling beam keeps constant and the atomic system is operated at near conditions of coherent population trapping due to atomic coherence. The refractive and absorptive ${\chi }^{(3)}$ nonlinearities enhanced by atomic coherence provide the physical mechanism of the phenomena. Based on the effects, all-optical flip-flop and storage of optical pulse signals with a low peak power of several tens of microwatts are implemented.

We predict the existence of bright and dark gap solitons in a single slab of negative index material. The formation of gap solitons is made possible by the exceptional interplay between the linear dispersive properties of the negative index etalon and the effect of a cubic nonlinearity.

We report experimental results on the three-dimensional Lagrangian acceleration in highly turbulent flows. Tracer particles are tracked optically using four silicon strip detectors from high energy physics that provide high temporal and spatial resolution. The components of the acceleration are shown to be statistically dependent. The probability density function of the acceleration magnitude is comparable to a log-normal distribution. Assuming isotropy, a log-normal distribution of the magnitude can account for the observed dependency of the components. The time dynamics of the acceleration components is found to be typical of the dissipation scales, whereas the magnitude evolves over longer times, possibly close to the integral time scale.

Based on the invariance principle of differential equations a simple, systematic, and rigorous feedback scheme with the variable feedback strength is proposed to stabilize nonlinearly finite-dimensional chaotic systems without any prior analytical knowledge of the systems. Especially the method may be used to control near-nonhyperbolic chaotic systems, which, although arising naturally from models in astrophysics to those for neurobiology, all Ott-Grebogi-York type methods will fail to stabilize. The technique is successfully used for the famous Hindmarsh-Rose neuron model, the FitzHugh-Rinzel neuron model, and the Rössler hyperchaos system, respectively.

We report measurements of three-wave mixing amplitudes on systems whose third order elastic constants have also been measured by means of the elastoacoustic effect. Because attenuation and diffraction are an important aspect of our measurement technique we analyze our results using a modified Khoklhov-Zabolotskaya-Kuznetsov equation in the frequency domain. We find that the value of $\beta$ so deduced for polymethyl methacrylate agrees quite well with that predicted from the stress dependent sound speed measurements, establishing that polymethyl methacrylate may be considered as a hyperelastic solid. The $\beta$ values of sedimentary rocks, though they are typically 2 orders of magnitude larger than, e.g., polymethyl methacrylates, are still a factor 3–10 less than those predicted from the elastoacoustic effect.

We show that a wide class of uncoupled limit-cycle oscillators can be in-phase synchronized by common weak additive noise. An expression of the Lyapunov exponent is analytically derived to study the stability of the noise-driven synchronizing state. The result shows that such a synchronization can be achieved in a broad class of oscillators with little constraint on their intrinsic property. On the other hand, the leaky integrate-and-fire neuron oscillators do not belong to this class, generating intermittent phase slips according to a power law distribution of their intervals.

We consider the paraxial model for a nonlinear resonator with a saturable absorber beyond the mean-field limit. For accessible parametric domains we observe total radiation confinement and the formation of 3D localized bright structures. Different from freely propagating light bullets, here the self-organization proceeds from the resonator feedback, combined with diffraction and nonlinearity. Such “cavity” light bullets can be independently excited and erased by appropriate pulses, and once created, they endlessly travel the cavity round-trip.