An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is proposed. We prove that the steady state can never be stabilized if the system and controller in sum have an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize and track saddle points (as well as unstable foci) is demonstrated both numerically and experimentally with an electrochemical Ni dissolution system.

Light transport in a strongly scattering, strongly anisotropic material is studied experimentally using both static and time-resolved techniques. Both the static and the dynamic results are well characterized by a diffusion equation with an anisotropic diffusion tensor and a scalar absorption term. Light diffuses 4.0 times faster along the uniaxial axis of the material compared with diffusion in the orthogonal directions.

We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. We demonstrate that spreading of the droplet of tracers in such flows is anomalously fast. Since the flow is equivalent to the integrable Hamiltonian system with 1 degree of freedom, this provides an example of integrable dynamics with long-range correlations, fractal power spectrum, and anomalous transport properties.

We study the stability of localized structures in a passive optical bistable system. We show that there is a critical value of the input field intensity above which localized structures are unstable with respect to a curvature instability. Beyond this instability boundary, a transition from the localized branch of solutions to stable hexagons is found.

We have proposed theoretically and studied numerically the existence of discrete breathers (intrinsic localized modes) in the dynamics of a two-dimensional Josephson-junction array biased by radio-frequency fields. The solutions are linearly stable in the framework of the Floquet theory and robust in the presence of thermal fluctuations. We have also discussed the conditions for realizing an experimental detection of these modes.

Active and passive scalars transported by an incompressible two-dimensional conductive fluid are investigated. It is shown that a passive scalar displays a direct cascade towards the small scales while the active magnetic potential builds up large-scale structures in an inverse cascade process. Correlations between scalar input and particle trajectories are found to be responsible for those dramatic differences as well as for the behavior of dissipative anomalies.

Experiments in quasi-two-dimensional geometry (Hele-Shaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.

We show by both experiment and theory that a specific class of planar conducting fractals possesses a series of self-similar resonances, leading to multiple gaps and pass bands for electromagnetic waves over an ultrawide frequency range. A double stack of these fractal patterns exhibits polarization-independent absolute gaps over a wide range of incidence angles. These characteristics are retained even when the fractal patterns are significantly subwavelength in all dimensions. In addition, the transmittance can be modulated by an external current source.

We demonstrate experimentally that the near-field and far-field transverse patterns of a large aperture vertical cavity surface emitting laser (VCSEL) can be successfully interpreted as a two-dimensional (2D) billiard system. It is found that the near-field and far-field transverse patterns of a large aperture VCSEL evidently represent the coordinate-space and momentum-space wave functions of a 2D quantum billiard, respectively. The result of this paper suggests that large aperture VCSELs are potentially appropriate physical systems for the wave-function study in quantum problems.

It has long been recognized that kinetic friction $F_k$ between two solids must be due to instabilities, sudden “pops” of certain degrees of freedom. Here, such pops are studied with a focus on boundary lubrication. The pops’ characteristics and consequently the friction-velocity relationship depend qualitatively on dimensionality, commensurability, and details of the lubricant wall interaction. It is found that $F_k$ should be small between commensurate surfaces. $F_k$ is large for incommensurate surfaces, unless the lubricant’s motion is confined to 1D. The effects of thermal noise are discussed and computer simulations are employed to show the relevance of the predictions to less idealized models.

We experimentally show that the response of a quantum-chaotic system can display resonance lines sharper than the inverse of the excitation duration. This allows us to discriminate two neighboring frequencies with a resolution nearly 40 times better than the limit set by the Fourier inequality. Furthermore, numerical studies indicate that there is no limit, but the loss of signal, to this resolution, opening ways for the development of sub-Fourier quantum-chaotic signal processing.

In this paper we present the first results on emission in metamaterial. We show how the specific properties of metallic composite material can modify the emission of an embedded source. We show that under proper conditions the energy radiated by a source embedded in a slab of metamaterial will be concentrated in a narrow cone in the surrounding media. An experimental demonstration of this effect is given in the microwave domain, and the constructed antenna has a directivity equivalent to the best reported results with photonic-crystal-based antennas but using a completely different physical principle [B. Temelkuaran et al., J. Appl. Phys. 87, 603 (2000)].

A paradigmatic nonhyperbolic dynamical system exhibiting deterministic diffusion is the smooth nonlinear climbing sine map. We find that this map generates fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. The measure of these self-similar sets is positive, parameter dependent, and in case of normal diffusion it shows a fractal diffusion coefficient. By using a Green-Kubo formula we link these fractal structures to the nonlinear microscopic dynamics in terms of fractal Takagi-like functions.

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power-law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with $[1+(q-1)(t/\tau {)}^2{]}^{1/(1-q)}$ (with the nonextensive entropic index $q$ and $\tau$ depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.

A bidisperse granular gas in a compartmentalized system is experimentally found to cluster competitively: Depending on the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the one containing mainly large particles. The experimental observations are quantitatively explained within a flux model.

The combination of several high order harmonics can produce an attosecond pulse train, provided that the harmonics are locked in phase to each other. We present calculations that evaluate the degree of phase locking that is achieved in argon and neon gases interacting with an intense, 50 fs laser pulse, for a range of macroscopic conditions. We find that phase locking depends on both the temporal and the spatial phase behavior of the harmonics, as determined by the interplay between the intrinsic dipole phase and the phase matching in the nonlinear medium. We show that, as a consequence of this, it is not possible to compensate for a lack of phase locking by purely temporal phase manipulation.

We investigate numerically the transition between static equilibrium and dynamic surface flow of a 2D cohesionless granular system driven by a continuous gravity loading. This transition is characterized by intermittent local dynamic rearrangements and can be described by an order parameter defined as the density of critical contacts, i.e., contacts where the friction is fully mobilized. Analysis of the spatial correlations of critical contacts shows the occurrence of “fluidized” clusters which exhibit a power-law divergence in size at the approach of the stability limit. The results are compatible with recent models that describe the granular system during the static/dynamic transition as a multiphase system.

We study dynamics of freely cooling granular gases in two dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, $E(t)\sim t^{-1}$, and velocity statistics are characterized by a universal Gaussian distribution in the long time limit. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, $\Delta v$, are small compared with the typical velocity, $\Delta v/v\sim t^{-1/4}$. Furthermore, locally averaged shear modes dominate over acoustic modes. The small thermal velocity fluctuations suggest that the system can be heuristically described by Burgers-like equations.

We report the observation of depression solitary surface waves on a layer of mercury when its depth is thin enough compared to the capillary length. These waves, as well as the well known elevation solitary waves, are studied with a new measurement technique using inductive sensors. The shape of the solitary waves, their amplitude-dependent velocity, and their damping rates by viscosity are found in good agreement with theoretical predictions.

We investigate a new type of quantum ratchet which may be realized by cold atoms in a double-well optical lattice, pulsed with unequal periods. The classical dynamics is chaotic and we find the classical diffusion rate $D$ is asymmetric in momentum up to a finite time $t_r$. The quantum behavior produces a corresponding asymmetry in the momentum distribution which is “frozen-in” by dynamical localization provided the break time $t^{*}\ge t_r$. We conclude that the cold atom ratchets require $Db/\hslash \sim 1$, where $b$ is a small deviation from period-one pulses.