Tunneling phenomena in the presence of chaos are investigated. Several remarkable features, which form a sharp contrast to the tunneling in the integrable system, can be well understood in terms of the semiclassical theory including complex classical trajectories. In particular, it is found that the dominant contribution to the tunneling regime comes from many complex branches which are not connected with any real manifolds and are linked at caustics to form the bifurcation chains. Chaotic tunneling is proposed as a new class of tunneling phenomena which originate from the complicated nature of the complex dynamical system.

Dynamical localization in general two-sided kicked rotors, which are classically nonintegrable, is shown to occur in the immediate vicinity of quantum antiresonance (periodic recurrences). A complete and exact solution of the quasienergy eigenvalue problem is obtained for the standard potential. Numerical evidence is given that this solution is an excellent approximation to the quantum dynamics and quasienergy states even not very close to antiresonance. The dynamical problem is mapped into a tight-binding model of a two-channel strip with pseudorandom disorder. One then has strong evidence for Anderson localization in this model near antiresonance.

A 2D model is introduced to study the onset of parametric surface waves, their secondary instabilities, and the transition to spatiotemporal chaos. We obtain the stability boundary of a periodic standing wave above onset against Eckhaus, zigzag, and transverse amplitude modulations (TAM), as a function of the control parameter $\varepsilon$ and the wavelength of the pattern. The Eckhaus and TAM boundaries cross at a finite value of $\varepsilon$, thus explaining the finite threshold for the TAM observed experimentally. At larger values of $\varepsilon$, a numerical solution reveals a transition to spatiotemporal chaotic states mediated by the TAM instability.

We study theoretically and by simulation the spatial correlation of the fluctuations of the transversely averaged fraction of the invading phase in invasion percolation. For an uncorrelated lattice, the profile has the correlation structure of the trace of fractional Brownian motion (FBM) with Hurst exponent $H=D-2\mathrm{}$ for 3D percolation and $H=D-1.5\mathrm{}$ for 2D percolation, where $D$ is the fractal dimension of the percolation cluster. Extension to correlated percolation shows similar results, with the FBM signal displaying persistent correlations ($H>\mathrm{}0.5\mathrm{}$) in 3D and antipersistent correlations ($H<\mathrm{}0.5\mathrm{}$) in 2D percolation. Experiments of nonwetting fluid invasion in porous media confirm the theory in 3D.

The first results on statistical properties of quantum quasidegeneracy are presented. A physical interpretation of the Shnirelman theorem which predicted bulk quasidegeneracy is given. Conditions for the strong impact of degeneracy on quantum level statistics are formulated, which allow us to extend the application of the Shnirelman theorem to a broad class of quantum systems.

For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by random matrix theory. We present here two counterexamples—the hydrogen atom in a magnetic field and the quartic oscillator—which display nearest neighbor statistics strongly different from the usual Wigner distribution. We interpret the results with a simple model using a set of regular states coupled to a set of chaotic states modeled by a random matrix.

We present a new theoretical and numerical framework for the study of the optical properties of micrometric and nanometric three-dimensional structures of arbitrary shape. We show that the field distribution induced inside and outside such a structure by different external monochromatic sources can be obtained from a unique generalized field propagator expressed in direct space. An application of the method to the confinement of optical fields due to the scattering of subwavelength objects is presented.

Defect motion in pattern formation (gliding and climbing, defects annihilation, pinning effects, and pattern reorientation) is analyzed for the first time when a fluid layer heated from below rotates slowly around its vertical axis. This system allows for a continuous change from a (approximately) variational problem (no rotation) to a nonvariational one by increasing the externally applied rotation rate. By a numerical integration of the 3D-hydrodynamic equations, climbing and gliding velocities of defects are determined. A mechanism leading to a reorientation of a parallel roll structure is discussed.

The nonlinear evolution of large structure in Rayleigh-Taylor and Richtmyer-Meshkov bubble and spike fronts is studied numerically and explained theoretically on the basis of single-mode and two-bubble interaction physics at Atwood numbers ($A$). Multimode Rayleigh-Taylor bubble (spike) fronts are found as $h_B={\alpha }_B\mathrm{Ag}{t}^2$ [$h_s={\alpha }_s\mathrm{}\left(A\right)g{t}^2$] with ${\alpha }_B=0.05\mathrm{}$, while Richtmyer-Meshkov bubble (spike) fronts are found as $h_B=a_B{t}^{{\theta }_B}$ ($h_s=a_s{t}^{{\theta }_s\mathrm{}\left(A\right)\mathrm{}}$) with ${\theta }_B=0.4\mathrm{}$ at all $A\text{'}\mathrm{s}$. The dependence of these scaling laws and parameters on $A$ and on initial conditions is explained.

We study the instability induced by the gravity of a liquid layer hanging at the bottom of a horizontal cylinder and continuously supplied with "fresh" liquid. Depending on the flow rate, different regimes are observed: dripping, array of liquid columns, and liquid sheets. The column array exhibits collective behaviors that can be forced under well controlled conditions by driving the displacements of two boundary columns: optical mode (regular or "chaotic"), phase diffusion, and propagation of localized "dilation waves" of the pattern.

A general method to describe Hamiltonian chaos in the thermodynamic limit is presented which is based on a model equation independent of the dynamics. This equation is derived from a geometric approach to Hamiltonian chaos recently proposed, and provides an analytic estimate of the largest Lyapunov exponent $\lambda$. The particular case of the Fermi-Pasta-Ulam $\beta$-model Hamiltonian is considered, showing an excellent agreement between the values of $\lambda$ predicted by the model and those obtained with computer simulations of the tangent dynamics.

We analyze the damped driven discrete sine-Gordon equation. For underdamped, highly discrete systems, we show that whirling periodic solutions undergo parametric instabilities at certain drive strengths. The theory predicts novel resonant steps in the current-voltage characteristics of discrete Josephson rings, occurring in the return path of the subgap region. We have observed these steps experimentally in a ring of 8 underdamped junctions. An unusual prediction, verified experimentally, is that such steps occur even if there are no vortices in the ring. Numerical simulations indicate that complex spatiotemporal behavior occurs past the onset of instability.

We calculate the magnetic response of ensembles of small two-dimensional structures at finite temperatures. Using semiclassical methods and numerical calculation we demonstrate that only short classical trajectories are relevant. The magnetic susceptibility is enhanced in regular systems, where these trajectories appear in families. For ensembles of squares we obtain a large paramagnetic susceptibility, in good agreement with recent measurements in the ballistic regime.

We consider chains of one-dimensional, piecewise linear, chaotic maps with uniform slope. We study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the Frobenius-Perron equation. For Markov partition values of the slope, we relate the diffusion coefficient to eigenvalues of the topological transition matrix. The diffusion coefficient obtained shows a fractal structure as a function of the slope of the map. This result may be typical for a wide class of maps, such as two-dimensional sawtooth maps.

We report a distinct transition at $\epsilon \approx 0.55$ from a state with dislocation defects to spiral-defect chaos in a Rayleigh-Bénard system of aspect ratio $\Gamma =40\mathrm{}$. The transition occurred over a narrow range in $\epsilon$ and was seen in angle-time representations of azimuthal power spectra, in the spatially averaged roll curvature, in the skewness of the structure function, and in the roll-nucleation frequency of sidewall foci. Comparison with work by others shows that the onset of spiral-defect chaos moves to smaller $\epsilon$ as $\Gamma$ increases.

Quantum steady state processes far from equilibrium can be characterized by wave functions whose coefficients (in a fixed basis) evolve onto a fractal set. This is explicitly shown for the prototypical situation of heat flow from hot to cold reservoirs, with two spin-½ particles in a thermal gradient. The resulting dimensional loss, induced by the nonequilibrium environment, is related to the quantum transport process.

A uniform semiclassical propagator is used to time evolve a wave packet in a smooth Hamiltonian system at energies for which the underlying classical motion is chaotic. The propagated wave packet is Fourier transformed to yield a scarred eigenstate.

We present a noise-free and deterministic dynamic model to study the dynamics of the contact line of an adhesive tape during the peeling process. The adhesive tape is wrapped around a disk and is pulled with a constant speed at a fixed distance. In a narrow range of the pull speed, the angular speed of the contact line exhibits stick-slip behaviors with irregular bifurcations, which is then followed by the appearance of a period one stick-slip motion before displaying chaotic behaviors with distinctively positive Liapunov exponents.

We report the observation of two-dimensional periodic and quasiperiodic structures in the transverse profile of an optical beam circulating in a loop which contains a nonlinear medium. The symmetries of the patterns are imposed by an image rotation within the loop. We propose a simple model which shows the existence of two unstable bands of transverse wave vectors. Close to threshold, the predictions of the model are qualitatively and quantitatively confirmed by the experiment.

It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a log-infinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems.