This paper discusses a 2D vortex merger for the particular case of a weak point vortex and an extended vortex of nearly circular cross section. A Hamiltonian analysis describes the interaction of the point vortex with surface waves on the extended vortex. Critical layers around the extended vortex are introduced where “point vortex–surface wave” resonances occur. Vortex merger occurs only when there is an overlap of neighboring critical layers. In this case the point vortex cascades through a sequence of resonances as it spirals in and merges with the extended vortex.

This experiment examines the stability of a smectic-A liquid-crystal film subject to a uniform shear flow with velocity and velocity gradient in the plane of the smectic layers. The experiment is made possible by virtue of a freely suspended liquid-crystal film which is sheared at its edge by a Couette apparatus. It was found that while there was no shear-induced phase transition, line defects were nucleated in the film when the shear rate $\dot{\gamma }$ attained a certain threshold. The critical shear rate ${\dot{\gamma }}_c$ was measured as a function of temperature and film thickness.

We show that a purely classical phase lag (or advance) is associated with homoclinic motion near a periodic orbit. Trajectories which leave the separatrix region belonging to a given unstable periodic orbit return with a phase shift relative to the periodic orbit, at fixed energy. The phase is topological in that it counts the number of homoclinic cycles but is insensitive to how long it takes to make them.

The inertial range scalings of local averages of energy dissipation rate and enstrophy (vorticity squared) are studied using high resolution direct numerical simulation data for homogeneous and isotropic turbulence. The Taylor microscale Reynolds number is 216. It is found that the enstrophy is more intermittent than dissipation, consistent with previous one-dimensional surrogate measurements at high Reynolds numbers. Contrary to some recent expectations, enstrophy and dissipation have different exponents.

Statistical properties of random distribution of strained vortices (Burgers' vortices) in turbulence are studied, and the scaling behavior of structure functions is investigated. It is found within the scale range of interest (corresponding to the inertial range) that the third-order structure function is negative and the scaling exponent is nearly unity in accordance with Kolmogorov's four-fifths law. The inertial-range scaling exponents are estimated up to the 25th order, which are in good agreement with those obtained from experiments and direct numerical simulations once the probability distribution of the vortex strength is taken into account.

A simple generalization of the Swift-Hohenberg equation is proposed as a model for the pattern-forming dynamics of a two-dimensional field with two unstable length scales. The equation is used to study the dynamics of surface waves in a fluid driven by a linear combination of two frequencies. The model exhibits steady-state solutions with 2-, 4-, 6-, and 12-fold symmetric patterns, similar to the periodic and quasiperiodic patterns observed in recent experiments.

The path of a circular air bubble rising in a viscous fluid of a Hele-Shaw cell is found to change from a straight path to a zigzag path when the Reynolds number of the bubble (proportional to the bubble terminal velocity) exceeds a threshold. By visualizing the wake structures of the rising bubbles, and measuring the actual paths of the bubbles, we demonstrate that the path instability is a consequence of the vortex shedding behind the bubble.

A Comment on the Letter by Marco Bianucci, Riccardo Mannella, and Paolo Grigolini, Phys. Rev. Lett. 77, 1258 (1996).

A Reply to the Comment by Massimo Falcioni and Angelo Vulpiani.

In the work we investigate the bifurcational mechanism of the loss of stability of the synchronous chaotic regime in coupled identical systems. We show that loss of synchronization is a result of the sequence of soft bifurcations of saddle periodic orbits which induce the bubbling and riddling transitions in the system. A bifurcation of a saddle periodic orbit embedded in the chaotic attractor determines the bubbling transition. The phenomenon of riddled basins occurs through a bifurcation of a periodic orbit located outside the symmetric subspace.

There are few examples in dynamical systems theory which lend themselves to exact computations of macroscopic variables of interest. One such variable is the Lyapunov exponent which measures the average attraction of an invariant set. This article presents a family of noninvertible transformations of the plane for which such computations are possible. This model sheds additional insight into the notion of what it can mean for an attracting invariant set to have a riddled basin of attraction.

We calculate complete quasienergy spectra (rather than partial information thereon) from classical periodic orbits for the kicked top, throughout the transition from integrability to well-developed chaos. The standard error incurred for the quasienergies is a small percentage of their mean spacing, even though the effective Planck constant is not pushed to small values. The price paid is the inclusion of collective contributions of clusters of periodic orbits near bifurcations into Gutzwiller's trace formula.

Based on very accurate measurements performed on a superconducting microwave resonator shaped like a desymmetrized three-dimensional Sinai billiard, we investigate for the first time spectral properties of the vectorial Helmholtz, i.e., nonquantum wave equation for a classically totally chaotic and theoretically precisely studied system. We are thereby able to generalize some aspects of quantum chaos and present some results which are consequences of the polarization features of the electromagnetic waves.

In principle, the state space of a chaotic attractor can be partially or wholly reconstructed from interspike intervals recorded from experiment. We demonstrate the existence of chaotic stochastic resonance, wherein the state information content of a partial reconstruction, as measured by the predictability of the output spike train, can be increased by adding noise to the spike creation process.

A method for controlling low-order chaotic behavior of continuous spatiotemporal systems is developed and demonstrated in a complex, realistic 3D partial differential equation model that is used successfully for predicting El Niño events in the equatorial Pacific. An unstable periodic orbit that involves a full-domain oscillation is stabilized using a feedback control applied to a single degree of freedom at a carefully chosen single “choke point” in space. A general criterion is presented for determining the optimal points in reconstructed delay-coordinate phase space at which to apply the feedback control.

We present a simple viscous theory of free-surface flows in boundary layers, which can accommodate regions of separated flow. In particular, this yields the structure of stationary hydraulic jumps, both in their circular and linear versions, as well as structures moving with a constant speed. Finally, we show how the fundamental hydraulic concepts of subcritical and supercritical flow, originating from inviscid theory, emerge at intermediate length scales in our model.

Semiclassical eigenenergies and resonances are obtained from classical periodic orbits by harmonic inversion of Gutzwiller's semiclassical recurrence function, i.e., the trace of the propagator. Applications to the chaotic three disk scattering system and, as a mathematical model, to the Riemann zeta function demonstrate the power of the technique. The method does not depend on the existence of a symbolic code and might be a tool for a semiclassical quantization of systems with nonhyperbolic or mixed regular-chaotic dynamics as well.

A stochastic model, validated by numerical simulations, is used to analyze granular convection. The vertical flow field is found to depend on the hyperbolic cosine of the horizontal coordinate and on a mixed linear-exponential function of the vertical coordinate. Separate cellular automata and particle dynamic simulations are included. Results agree with existing experimental studies and give predictions to be tested experimentally.

Horizontal vibration of a granular material is studied experimentally and by means of molecular dynamics simulations. We focus on the transition from a more solidlike behavior of the granulate, which is obtained at small driving velocities, to a more fluidlike behavior for higher velocities. We present two different order parameters suitable to characterize the transition, namely the granular temperature and the pair distribution function. When using different frequencies $f$ and amplitudes $A_0$ of the vibration, we find that the transition point from the solid to the fluid phase scales with $A_0{f}^2$, whereas the granular temperature scales as $(A_{0}f{)}^{1.66}$.

The natural measure of a chaotic set in a phase-space region can be related to the dynamical properties of the unstable periodic orbits embedded in that set. This result has been proven to be valid for hyperbolic chaotic systems. We test the goodness of such a periodic-orbit characterization of the natural measure for nonhyperbolic chaotic systems by comparing the natural measure of a typical chaotic trajectory with that computed from unstable periodic orbits. Our results suggest that the unstable periodic-orbit formulation of the natural measure is typically valid for nonhyperbolic chaotic systems.