Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.

Experiments and computations indicate that mixing in chaotic flows generates certain coherent spatial structures. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifold of some periodic orbit) then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We demonstrate an amazing property for certain global structures: A basin has a basin cell if and only if every diverging curve comes close to every basin boundary point of that basin.

The set of initial conditions leading to transient chaos in the neighborhood of a crisis is shown to display riddled-like behavior for finite, arbitrarily small accuracy. Calculations are performed with the logistic equation, as well as with an experimentally verifiable, quantitative description of a chemical reaction. Limitations in computational or experimental accuracy make the apparent riddling of initial conditions a phenomenon that is indistinguishable from riddled basins.

We consider interaction of vortices in the vector complex Ginzburg-Landau equation (CVGLE). In the limit of small field coupling, it is found analytically that the interaction between well-separated defects in two different fields is long ranged, in contrast to the interaction between defects in the same field which falls off exponentially. In a certain region of parameters of CVGLE, we find stable rotating bound states of two defects—a “vortex molecule.”

The failure of a two-dimensional packing of elastic grains is analyzed using a numerical model. The packing fails through formation of shear bands or faults. During failure there is a separation of the system into two grain-packing states. In a shear band, local “rotating bearings” are spontaneously formed. The bearing state is favored in a shear band because it has a low stiffness against shearing. The “seismic activity” distribution in the packing has the same characteristics as that of the earthquake distribution in tectonic faults. The directions of the principal stresses in a bearing are reminiscent of those found at the San Andreas Fault.

We analyze the steady planar shear flow of the modified Johnson-Segalman model, which has an added nonlocal term. We find that the new term allows for unambiguous selection of the stress at which two “phases” coexist, in contrast to the original model. For general differential constitutive models we show the singular nature of stress selection in terms of a saddle connection between fixed points in the equivalent dynamical system. The result means that stress selection is unique under most conditions for space nonlocal models. Finally, illustrated by simple models, we show that stress selection generally depends on the form of the nonlocal terms (weak universality).

Let $T_p$ be the structure function of order $p$ of turbulence. First $T_2$ is determined, then $T_3$ is derived from $T_2$, and finally $T_p$ $(p>\mathrm{}3\mathrm{})$ are derived from $T_2$ and $T_3$. This closure scheme is realized by a non-Gaussian statistical model. We use it to study the scaling law of $T_p$, and we find that the available data on scaling exponents favor Kolmogorov's 1941 theory rather than his 1962 theory. We also predict the high-order universal constants of inertial-range scaling.

The statistics of uncorrelated point vortices in a plane is studied analytically and numerically. Theoretical distributions are obtained with the general method developed by Holtsmark [Ann. Phys. 58, 577 (1919)] and Chandrasekhar [Rev. Mod. Phys. 15, 1 (1943)]. They are found to agree with the results of numerical tests. Randomly placed Euler vortices have nearly Gaussian velocity distributions and Lorentzian distributions of the velocity difference. Statistics of other types of point vortices is essentially non-Gaussian.

Two-mode rhomboid patterns are generated experimentally via two-frequency parametric forcing of surface waves. These patterns are formed by the simple nonlinear resonance: ${\overrightarrow{k}}_2^{\prime }-{\overrightarrow{k}}_2={\overrightarrow{k}}_1$ where $k_1$ and $k_2\left({=k}_2^{\prime }\right)$ are concurrently excited eigenmodes. The state possesses a direction-dependent time dependence described by a superposition of the two modes, and a dimensionless scaling delineating the state's region of existence is presented. We also show that $2n$-fold quasipatterns naturally arise from these states when the coupling angle between ${\overrightarrow{k}}_2$ and ${\overrightarrow{k}}_2^{\prime }$ is $2\pi /n$.

We show that soliton excitation in intracavity down-conversion naturally selects a strictly defined frequency difference between the signal and idler fields. In particular, this phenomenon implies that if the signal has smaller losses than the idler then its frequency is pulled away from the cavity resonance and the idler frequency is pulled towards the resonance and vice versa. The frequency selection is shown to be closely linked with the relative energy balance between the idler and signal fields.

We show that modulation instability can exist with partially spatially incoherent light beams in a noninstantaneous nonlinear environment. For such incoherent modulation instability to occur, the value of the nonlinearity has to exceed a threshold imposed by the degree of spatial coherence.

The passive scalar convection by chaotic two-dimensional incompressible flow is studied. Analytically solvable equations are suggested to describe the evolution of the probability density functions of tracer gradients and power spectra. The parameters of the model are expressed explicitly via the correlation functions of the velocity field. The multifractal spectrum $f\left(\alpha \right)$ of the scalar dissipation field is calculated; strict multifractality holds only for small values of $\alpha$. Stationary and exponentially decaying power spectra of the scalar are obtained.

The scaling properties of three-dimensional magnetohydrodynamic turbulence with finite magnetic helicity are obtained from direct numerical simulations using ${512}^3$ modes. The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology. The scaling exponents of the structure functions can be described by a modified She-Leveque model ${\zeta }_p=p/9+1-(1/3{)}^{p/3}$, corresponding to basic Kolmogorov scaling and sheetlike dissipative structures. In particular, we find ${\zeta }_2\simeq 0.7$, consistent with the energy spectrum $E_k\sim k^{-5\mathrm{}/3}$ as observed in the solar wind, and ${\zeta }_3\simeq 1$, confirming a recent analytical result.

Experimental observations of rotating spiral flame edges formed during near-limit combustion of a downward-facing, polymethylmethacrylate disk spinning in quiescent air are reported. These flames exhibit similarities to patterns commonly found in quiescent excitable media. The tail rotates rigidly while the tip executes a compound, meandering motion sometimes observed in Belousov-Zhabotinskii reactions. A model assuming a rigid-body rotation with a constant speed of propagation relative to the swirling gas flow generated by the spinning disk predicts the observed spiral shapes well.

We address the problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type. We take into account the inertial term in order to understand the role of the chaotic dynamics in the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a periodic regime. Close to this bifurcation, we observed trajectories revealing intermittent chaos and anomalous deterministic diffusion.

We consider a billiard model of a self-bound, interacting three-body system in two spatial dimensions. Numerical studies show that the classical dynamics is chaotic. The corresponding quantum system displays spectral fluctuations that exhibit small deviations from random matrix theory predictions. These can be understood in terms of a new type of scarring caused by a one-parameter family of orbits inside the collinear manifold.

We derive the formalism to obtain spatial distributions of collisional correlation times for macroscopic particles undergoing granular flow from pulsed gradient spin echo nuclear magnetic resonance diffusion data. This is demonstrated with an example of axial motion in the shear flow regime of a 3D granular flow in a horizontal rotating cylinder at one rotation rate.

A Comment on the Letter by Y. Nahmad-Molinari, C. A. Arancibia-Bulnes, and J. C. Ruiz-Suárez, Phys. Rev. Lett. 82, 727 (1999). The authors of the Letter offer a Reply.

A Reply to the Comment by David Linton Johnson.