A control scheme is devised for taming the dynamics of convectively unstable extended systems exhibiting chaotic behavior. We show how complex spatiotemporal phenomena can be eliminated in arrays of coupled chaotic elements, in favor of a coherent state in which all elements are synchronized to a prescribed periodic orbit of the uncoupled system. The propagation of the synchronizing front and the resulting steady state properties of the array are anlayzed in the presence of noise.

The level of variance maintained in a stochastically forced asymptotically stable linear dynamical system with a non-normal dynamical operator cannot be fully characterized by the decay rate of its normal modes, in contrast to the case of normal dynamical systems. The nonorthogonality of modes in a non-normal system may lead to transient growth which supports variance far in excess of that anticipated from the decay rate given by the eigenvalues of the operator. As an example, the variance maintained by stochastic forcing in a canonical shear flow is examined in this work and found to increase with a power of the Reynolds number between 1.5 and 3. This great amplification of variance suggests a fundamentally linear mechanism underlying shear flow turbulence.

Geometric phases of real wave functions in nonintegrable quantum billiards are measured using microwave resonators. They appear as a sign change of the wave function after a cyclic excursion around a diabolic point in the space of shapes of the resonator. For a path encircling a pointlike triple degeneracy of an integrable system, an unexpected sign change for two out of the three wave functions is found, which must be attributed to a hidden symmetry, and not to a Berry phase.

In a random Gaussian wave field there is on average one vortex (phase singularity) for every two coherence areas. These vortices exhibit an unexpected, highly surprising correlation–nearest neighbors are 90% anticorrelated in sign. We show that on the zero crossings of the real or imaginary parts of the field adjacent vortices must be of opposite sign. This principle, which is unaffected by boundaries, accounts for the observed sign anticorrelations. We also show how the sign of any single vortex in a random wave field determines the sign of all other vortices in the wave field.

A generalization of self-guided beams (spatial solitons) is discussed. Each beam is characterized by an axially uniform intensity profile whose polarization state changes with propagation. The soliton therefore has internal dynamics. It is composed of two orthogonally polarized modes of the linear waveguide induced by the dynamic soliton which, in general, do not exist as solitons independently. Our insight and exact analytical results for the threshold nonlinearity are found from the elementary physics of linear optical waveguides via a nonlinear self-consistency procedure.

Solenoidal advection in d dimensions may induce inertial-range behavior 〈‖T(x,t)-T(x+r,t)${\mathrm{\Vert }}^{\mathit{n}}$〉∝${\mathit{r}}^{\mathrm{\zeta }\mathit{n}\left(\mathit{T}\right)}$ of a passive scalar field T(x,t). If the velocity field changes very rapidly in time, the ${\mathrm{\zeta }}_{\mathit{n}}$(T) are determined by the two-particle eddy-diffusion coefficient η(r)∝${\mathit{r}}^{\mathrm{\zeta }\left(\mathrm{\eta }\right)}$. An approximation to the molecular-diffusion terms yields ${\mathrm{\zeta }}_{\mathit{n}}$(T)=1/2 √2nd[2-ζ(η)]+${\mathrm{\alpha }}^2$ -1/2α where α=d+ζ(η)-2. As n→∞, ${\mathrm{\zeta }}_{\mathit{n}}$(T)∝ √nd .

We have measured the optical transmission of quasiperiodic dielectric multilayer stacks of ${\mathrm{SiO}}_2$ (A) and ${\mathrm{TiO}}_2$ (b) thin films which are ordered according to a Fibonacci sequence ${\mathit{S}}_{\mathit{j}+1\mathrm{}}$={${\mathit{S}}_{\mathit{j}\mathrm{-}1}$}, with ${\mathit{S}}_0$={B} and ${\mathit{S}}_1$={A} up to the sequence ${\mathit{S}}_9$ which consists of 55 layers. We observe a scaling of the transmission coefficient with increasing Fibonacci sequences at quarter-wavelength optical thicknesses. This behavior is in good agreement with theory and can be considered as experimental evidence for the localization of the light waves. The persistence of strong suppression of the transmission (gaps) in the presence of variations in the refractive indices among the layers is surprising.

Time-reversal mirror (TRM) utilizes time-reversal invariance of wave equation. Acoustic TRM are made of large transducer arrays allowing the incident acoustic field to be sampled, time reversed, and reemitted. The ability of TRM to time reverse separately each scattered wave permits us to isolate spatially and temporally the equivalent sources of extended scatterer illuminated by broadband pulse. We investigate more particulary the problem of a cylindrical scatterer: We show that TRM allows us to determine the location of the mode conversions of incident waves into circumferential waves.

We report imbibition experiments investigating the effect of evaporation on the interface roughness and mean interface height. We observe a new exponent characterizing the scaling of the saturated surface width. Further, we argue that evaporation can be usefully modeled by introducing a gradient in the strength of the disorder, in analogy with the gradient percolation model of Sapoval et al. By incorporating this gradient we predict a new critical exponent and a novel scaling relation for the interface width. Both the exponent value and the form of the scaling agree with the experimental results.

We discuss time-dependent spatially localized solutions of the quintic complex Ginzburg-Landau equation applicable near a weakly inverted bifurcation to traveling waves. We find that there are—in addition to the stationary pulses reported previously—stable localized solutions that are periodic, quasiperiodic, or even chaotic in time. An intuitive picture for the stability of these time-dependent localized solutions is presented and the novelty of these phenomena in comparison to localized solutions arising for exactly integrable systems is emphasized.

The behavior of two antiparallel vortices of equal strength, moving in an ideal fluid, is studied numerically. The system is unstable and after a transient period two points of the vortices collide. On their way to the collision the two vortices form a pyramidal structure which is independent of the initial conditions. The connection process tends to follow a universal route for all kinds of initial vortex-antivortex arrangements.

Three complex ordinary differential equations, modeling interactions in wall-bounded turbulent flow, are developed and studied. It is believed that the roll-wave interactions which are included are of fundamental importance to turbulence production. Integration of these triad equations is shown to be consistent with direct flow simulations and the Ruelle-Takens route to chaos.

The Kolmogorov relation for the third order structure function is used to derive the energy spectrum in the far dissipation range (k→∞). This contains no unspecified constants. Using methods from matched asymptotic expansions and mild analyticity assumptions, a uniformly valid form for the inertial through the dissipative ranges is obtained. An analogous energy spectrum is presented. This is compared with the results of physical and numerical experiments on the energy spectra E(k). The theoretical predictions are found to deviate by not more than a few percent from the measured data in the entire range of wave numbers where the energy spectrum E(k) varies by more than 30 orders of magnitude.

We present a rigorous method based on the implicit function theorem for computing anomalous exponents for self-similar solutions to a variety of problems in continuum mechanics. Unlike formal perturbation methods which give only local results, our method often gives global results and analytic dependence of the exponent on parameters. We describe in detail the application of the method to source-type solutions to Barenblatt’s equation for elastoplastic flow, a problem which has been treated elsewhere by renormalization methods. We also briefly describe several other applications.

Experiments on double diffusive convection sustained by imposed vertical concentration gradients have been conducted in a novel thin isothermal parallelepiped cell. The measured critical Rayleigh number and the frequency of traveling waves are in reasonable accord with a stability analysis that predicts a tricritical condition all along the instability boundary. Visualization of the pattern from the side reveals that the asymptotic state immediately above the onset of instability consists of packets of traveling finite amplitude plumes.

Recent studies of universal parametric correlations in quantum chaotic spectra have revealed an astonishing connection to an integrable one-dimensional quantum system. We introduce a continuous matrix model which establishes a direct connection between the quantum Hamiltonian and the exact field theoretic description of spectral correlations. This reveals a common mathematical structure which underlies quantum chaos, matrix models, and a quantum Hamiltonian.

A spatiotemporal system is modeled by a coupled map lattice. Feedback pinnings are used to control chaos of the system by stabilizing a certain unstable reference state. As the pinning distribution is dense enough, the unstable state can be stabilized. If the density is low, the reference state may not be approached asymptotically. In this case, however, the pinnings can still effectively suppress chaos and produce rich spatiotemporal structures. If a solution is locally stable while the transient process to this state is extremely long and chaotic, pinnings of very low density can well control the transient chaos.

We show that the presence of Kramers’ degeneracy plays an important role in determining the localization length of a dynamically localizing quantum chaotic system. Based on the existence of mappings between dynamically localizing systems and tight binding models of disordered strongly localizing solids we conjecture that the localization length is not a universal function of the symmetry class.

A solution to a given equation is structurally stable if it suffers only an infinitesimal change when the equation (not the solution) is perturbed infinitesimally. We have found that structural stability can be used as a velocity selection principle for propagating fronts. We give examples, using numerical and renormalization group methods.

We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.