Abstract: Angular-resolved photoelectron spectra from the fivefold surface of single-grain icosahedral ${\mathrm{Al}}_{70}$${\mathrm{Pd}}_{21.5}$${\mathrm{Mn}}_{8.5}$ exhibit a quasiperiodic dispersion of 300 meV at 2.3 eV binding energy. Low energy electron diffraction studies confirm quasicrystalline order at the surface. A distinct pseudogap feature is observed with a density of states near the Fermi level decreasing as a square-root power law.

Abstract: Delayed robot systems, even of low degree of freedom, can produce phenomena which are well understood in the theory of nonlinear dynamical systems, but hardly ever occur in simple mechanical models. To illustrate this, we analyze the delayed positioning of a single degree of freedom robot arm which leads to an infinite dimensional dynamical system. Restricting the dynamics to a four dimensional center manifold, we show that the system undergoes a codimension two Hopf bifurcation for an infinite set of parameter values. This provides a mechanism for the creation of two-tori in the phase space and gives a theoretical explantion for self-excited quasiperiodic oscillations of force controlled robots. We also compare our results with experimental data.

Abstract: The electronic motion in quasiperiodic systems (the Harper model, the Fibonacci chain, two- and three-dimensional Fibonacci quasilattices) is studied, in the framework of a tight-binding Hamiltonian. The spreading with time of the wavepacket is described in terms of the behaviour of the autocorrelation function C(t). It is found that, in all cases, C(t) approximately t- delta . For the Harper model with lambda <2, the motion of the electron is ballistic ( delta =1), which goes against a previous estimate of delta =0.84. We show that this discrepancy is due to the neglect of a logarithmic contribution in the scaling analysis. For the Harper model with lambda =2 and the Fibonacci chain, the motion is non-ballistic with 0< delta <1. For the higher-dimensional Fibonacci quasilattices, C(t) exhibits a transition from a ballistic to a non-ballistic behaviour, upon varying the modulation strength of the quasiperiodicity. The relation between C(t) and the fractal dimensions of the spectral measure is also studied.

Abstract: The influence of fluid elasticity on the onset of aperiodic or chaotic motion of an upper-convected Maxwellian fluid is examined in the context of the Rayleigh-B\'enard thermal convection problem. A truncated Fourier representation of the flow and temperature fields leads to a four-dimensional dynamical system that constitutes a generalization of the classical Lorenz system for Newtonian fluids. It is found that, to the order of the present truncation and above a critical value of the Deborah number ${\mathrm{De}}^{\mathit{c}}$, steady convection cannot set in, with the fluid becoming overstable instead. For De${\mathrm{De}}^{\mathit{c}}$, and even close to the Newtonian limit, the presence of fluid elasticity appears to alter significantly the circumstances leading to the onset of chaotic motion. Depending on the value of the Prandtl number, chaos is found to set in through the quasiperiodic route or period doubling. In general, fluid elasticity tends to destabilize the convective cell structure, precipitating the onset of chaos, at a Rayleigh number that may be well below that corresponding to Newtonian fluids.

Abstract: We study the density of states, the distribution of energy spacings, and the transmission coefficient of one-dimensional quasiperiodic Fibonacci and Thue-Morse systems. We consider arrays of \ensuremath{\delta} potentials with constant separation and two potential strengths, and tight-binding systems with constant nearest-neighbor couplings and two different on-site energies. The quasiperiodicity lies in the arrangement of the two possible values of either the potential strengths or the on-site energies. We analyze the fractal character of the energy spectra of these systems through their integrated density of states and fractal dimensionality. We study the average with respect to energy of the transmission coefficient, which turns out to be a good way to measure the regularity of the system.

Abstract: The dynamics of a Brillouin fiber ring laser operating on the two first Stokes components is studied both theoretically and experimentally. The emergence of the second order Stokes wave is described within the framework of a coherent five-wave model that generalizes the usual three-wave model. The laser steady states are analytically characterized and the dynamics is numerically studied. The laser emission is stable except at high pumping rates for which the system exhibits periodic and quasiperiodic instabilities. Experimental results are in good agreement with the theoretical predictions.

Abstract: The near resonant response of suspended elastic cables driven by harmonic, planar excitation is investigated experimentally. Measurements of large amplitude cable motions confirm previous theoretical predictions of fundamental classes of internally-resonant responses. For particular magnitudes of equilibrium curvature, strong modal interactions arise through isolated (two-mode) or simultaneous (three-mode) internal resonances. Four qualitatively different periodic responses are observed : (1) pure planar response, (2) 2 :1 internally resonant nonplanar response, (3) 1 :1 internally resonant nonplanar response, and (4) simultaneous, 2 :2 :1 internally resonant nonplanar response. Quasiperiodic responses are also observed.

Abstract: Calculations are presented for the local density of phonon states scrD(${\mathrm{\ensuremath{\omega}}}^{2}$), the lattice specific heat ${\mathit{C}}_{\mathit{v}}$, and the transmission coefficient of a particle through a layered quasicrystal whose ends are fixed. In this paper, scrD(${\mathrm{\ensuremath{\omega}}}^{2}$) is obtained using a systematic decimation of the equations of motion for the atoms on a chain (where the two-dimensional plane is simulated by an atom) with the use of real-space renormalization-group techniques. The renormalized spring coupling constants for the atomic arrangement with the silver and golden means become invariant after the first decimation. However, the copper mean arrangement only becomes invariant after two decimations. We analyze the effect of this behavior in calculating the limiting case of a periodic chain from the Fibonacci series. We compare the generalized Fibonacci lattice with a lattice whose coupling constants are arranged in the Thue-Morse sequence. For the Fibonacci lattices, there is a significant difference between the copper mean results for the low-frequency density of states and those for the gold and silver lattices. This difference leads to a significant change in the specific heat for the copper relative to the periodic lattice. The density of states for the Thue-Morse chain has a unique low-frequency behavior and this also leads to a significant change in its specific heat at low temperature compared with a periodic lattice of the same length.

Abstract: A new vibration absorbing device is introduced for large flexible structures. The phase-space of the experimental system is reconstructed via delay-coordinate embedding technique. Experimental dynamics indicate that the motion is predominantly quasiperiodic, confirming the existence ofinvariant tori. Within the quasiperiodic region, there are windows containing intricate webs of phase-locked periodic responses. The quasiperiodic and the phase-locked responses are clearly visualized on the cover of the torus. Increase in the amplitude ofexcitation results in distortion of the invariant torus due to the resonance overlap. Due to the resonance overlap, the return map extracted from the experimental data becomes noninvertible. Furthermore, a burst of frequencies appears on the Fourier spectrum. This scenario is similar to many experimental observations of hydrodynamical instabilities; the breakup of the tori in these experiments is related to the onset of turbulence.

Abstract: Nonlinear saturation of unstable solutions to the weakly relativistic, one-dimensional Zakharov equations is considered in this paper In order to perform the analysis, two quantities are introduced One of them, ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, is proportional to the initial energy of the high-frequency field, and the other is the basic wave vector of the low-frequency perturbing mode k=2\ensuremath{\pi}/L, with L as the length scale With these quantities it becomes possible to identify a number of regions on a ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ versus k parametric plane For very small values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, steady-state solutions become unstable when k is also very small In this case ion-acoustic dynamics is found to be unimportant and the system is numerically shown to be approximately integrable, even if k is below a critical value where the solutions are not simply periodic For larger values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ the unstable wave vectors also become larger and the ion-acoustic fluctuations turn into active dynamical modes of the system, driving a transition to chaos, which follows initial inverse pitchfork bifurcations The transition includes resonant and quasiperiodic features; separatrix crossing phenomena are also found The influence of relativistic terms on the chaotic dynamics is studied in the context of the Zakharov equations; it is shown that relativistic terms generally enhance the instabilities of the system, therefore anticipating the transition

Abstract: We prove that quasiperiodic tilings of IWk, obtained by the strip projection method when the linear embedding of IWk in lWZk has quadratic coefficients, always admit local rules with decorations. lieywords: Quasiperiodic tilings, cut method, local rules.

Abstract: Measurements with photoemission spectroscopy in the photon-energy range 40--140 eV have been used to determine the valence band of the stable icosahedral ${\mathrm{Al}}_{65}$${\mathrm{Cu}}_{20}$${\mathrm{Ru}}_{15}$ Resonant photoemission near the Ru 4p\ensuremath{\rightarrow}4d transition has been employed to show that the feature in the valence band with the maximum intensity at 13(1) eV below the Fermi level is predominantly of Ru 4d character This has been additionally verified by conducting the photoemission measurements in the constant-initial-state mode and by using the effect of the Cooper minimum in the photoionization cross section of the Ru 4d orbitals The valence-band feature with the maximum intensity at 38(1) eV below the Fermi level has been shown as being due mainly to the states of Cu 3d character The Ru 4d and Cu 3d empirical partial density of states have been determined from the photoemission spectra The decrease of intensity towards the Fermi level has been interpreted as being indicative of the presence of a theoretically predicted pseudogap around the Fermi level It has been indicated, however, that the Fermi cutoff also contributes to the observed intensity decrease It has been demonstrated that the energy resolution of the spectroscopic measurements performed so far on quasicrystals was not high enough to unambiguously determine the presence of such a pseudogap No unusual features in the valence band of icosahedral ${\mathrm{Al}}_{65}$${\mathrm{Cu}}_{20}$${\mathrm{Ru}}_{15}$, which could be ascribed to its quasiperiodic nature, have been observed within the resolution of the experiment The need of high-energy-resolution spectroscopic measurements, which are essential in order to observe the theoretically predicted spikiness of the density of states in quasicrystals, has been emphasized A review of published experimental data on the electronic structure of quasicrystals has also been presented

Abstract: A renormalization scheme which takes into account the natural frequency of the system is developed to study an anisotropic quantum XY spin chain in a quasiperiodic transverse field. The quasiparticle excitations of the model exhibit extended, localized as well as critical phase, with fractal characteristics, in a finite parameter interval. The scaling properties of the critical phase fall into four distinct universality classes. The isotropic limit of the model describes the extensively studied Harper equation. The renormalization approach provides a new method for determining energies and transition thresholds with extremely high precision.

Abstract: The asymptotic behavior of mean values for integrals of quasiperiodic functions, which characterizes the uniformity of the distribution of irrational windings on a torus, is shown to be essentially dependent on the dimension of the torus. We prove the nonrecurrence of mean values for arbitrarily smooth three-frequency quasiperiodic functions. We also present a series of results concerning the distribution of fractional parts for systems of linear functions.

Abstract: We consider quasiperiodic positive definite Lagrangian systems and establish the existence of one or more solutions homoclinic (namely asymptotic as $t\to\pm\infty$) to a quasiperiodic solution. These results are obtained by means of a variational approach. An application is carried out to quasiperiodically perturbed Lagrangian systems.

Abstract: The single Bonhoeffer-van der Pol (BvP) neuron, driven by a sinusodial external stimulus shows periodic, quasiperiodic and chaotic states. Frequency-lockings, followed by period doubling cascades to chaos, suggest a relation to the universal behaviour of one-dimensional circle maps. We present a method to describe the membrane potential activity of single neurons with a circle map. Furthermore, we show that this equivalence can be found in networks of coupled BvP elements. In dependence of the interaction strength we always observe a region, where a stimulus-induced synchronization can be understood in terms of coupled one-dimensional circle maps.

Abstract: By establishing the correspondence between the substitution rule (a→anb; b→a) and the transformation on the value of ω=tan φ byω→1/(n+ω) in the cut-and-project (CP) method, it is proved that the necessary and sufficient conditions for a binary quasiperiodic (QP) sequence made by the CP method to be self-similar is that ω is a quadratic irrational (QI) number. And, vice versa, the necessary condition for a binary self-similar sequence generated by the substitution rule to be obtainable by the CP method is that the corresponding substitution rule can be rewritten as a simple composition of transformations with the type (a→anb; b→a). To illustrate some physical properties of the self-similar QP chains associated with QI numbers, we analyze the scaling behaviour of the wave function atE=0 for an off-diagnonal tight-binding model. It is shown that the wave function atE=0 grows at most by a power-law for the QP lattices, characterized by a special class of QI numbers. For the QP chains associated with general QI numbers, with the same logic, the typical scaling behaviour of the wave function is conjectured to be the same.

Abstract: The Weierstrass function is continuous everywhere and differentiable nowhere. It is used advantageously for modeling fractally coarse media. The frequency domain expression is analyzed using the Laplace transform of a truncated Weierstrass function. The poles of this function constitute a geometric sequence on the imaginary axis of the complex frequency plane. Computer calculations show that the zeros change little as the order is increased. Theoretical development reveals the nature of the dependence of the zeros on order and the parameters of the function. The effect of the scale of observation on the function's properties is examined in both time and frequency domains.

Abstract: We present a detailed investigation on the decagonal quasicrystal (D-phase) formed from an Al-Pd-Mn icosahedral quasicrystal (I-phase) through a solid-state phase transformation, including its formation, compositional and crystallographical relationships with the matrix I-phase, growth mode, and structural characteristics. The as-melt-spun Al70Pd20Mn10 alloy contains only I-phase. By annealing at 800 °C, the D-phase is found to grow cpitaxially from the I-phase to establish a D/I two-phase equilibrium with distinctly different composition between them. The D-phase exhibits a stepped growth interface, which consists of a facet plane, formed by sharing the tenfold plane with a fivefold plane of the matrix I-phase, and some ledges across it. The growth of the D-phase into the I-phase proceeds through lateral movement of the ledges along the tenfold plane. High-resolution electron microscopy reveals that the structure of the D-phase is constructed by an aperiodic arrangement of decagonal atom clusters with definite linkages and long-range quasiperiodic correlation.

Abstract: Steady vortex flows over highly swept wings develop quasiperiodic velocity fluctuations. The nature of such fluctuations was explored extensively using a 59.3-deg cropped delta wing at 25-deg incidence in a low-speed wind tunnel. Additional single point tests were conducted over a range of incidences (16-32 deg) and Reynolds numbers (1.2 x 105 to 5.6 x 10 s) based on the root chord. Cross-spectral analysis of velocity fluctuations, sensed by two hot-wire sensors, was used to track these phenomena to the region of their origin as well as study the evolution and growth of the fluctuations. Results show the existence of narrow, dominant frequency bands containing the majority of the fluctuation energy. At 25 deg the quasiperiodicity originates near the 30% span region and the intensity amplifies downstream as the corresponding peak frequency decreases. Further downstream the frequency levels off while the intensity peaks, then decreases. Coherence trajectory mapping displays a helical path around the core of the vortex system. At a fixed location relative to the model, the product of the Strouhal number and the nominal wake scale was relatively constant with respect to freestream speed.

Abstract: A theoretical study of the effects of in‐plane magnetic fields on the interband optical absorption spectra of quasiperiodic GaAs–(Ga,Al)As Fibonacci superlattices is presented within the effective‐mass approximation. The electron‐envelope wave functions and magnetic subbands are obtained by an expansion in harmonic‐oscillator wave functions. The theoretical optical absorption spectra are calculated for magnetic fields related by integer powers of the golden mean τ=(1+√5)/2. It is unambiguously shown that, for magnetic‐field values scaled by τ2n, the corresponding optical absorption spectra essentially exhibit a self‐similar behavior, with the width of the peaks increasing linearly with the field, in agreement with the experimental results by D. Toet, M. Potemski, Y. Y. Wang, J. C. Maan, L. Tapter, and K. Ploog [Phys. Rev. Lett. 66, 2128 (1991)].

Abstract: A family of generalized Fibonacci lattices, which are locally isomorphic and exhibit a peculiar type of self-similarity, have been found to exist by shifting the position of the strip (initial phase) in the projection method. The necessary and sufficient conditions of generating such self-similar quasiperiodic lattices are established. The power-law growth behaviour of the wavefunction at E=0 in the off-diagonal model defined on some of these lattices has been analysed. It is shown that, although various structures resulting from different initial phases are in a local isomorphism class, they lead to a variety of maximum exponents of power for the scaling of the wavefunction.