Abstract: We experimentally investigate the structural behavior of an interacting colloidal monolayer being driven across a decagonal quasiperiodic potential landscape created by an optical interference pattern. When the direction of the driving force is varied, we observe the monolayer to be directionally locked on angles corresponding to the symmetry axes of the underlying potential. At such locking steps, we observe a dynamically ordered smectic phase in agreement with recent simulations. We demonstrate that such dynamical ordering is due to the interaction of particle lanes formed by interstitial and noninterstitial particles.

Abstract: A quasiperiodic intermediate-valence (IV) system is realized in an icosahedral Au-Al-Yb quasicrystal. X-ray absorption spectroscopy near the Yb ${L}_{3}$ edge indicates that quasiperiodically arranged Yb ions assume a mean valence of 2.61, between a divalent state ($4{f}^{14}$, $J=0$) and a trivalent one ($4{f}^{13}$, $J=7/2$). Magnetization measurements demonstrate that the $4f$ holes in this quasicrystal have a localized character. The magnetic susceptibility shows a Curie-Weiss behavior above $\ensuremath{\sim}100$ K with an effective magnetic moment of 3.81${\ensuremath{\mu}}_{\mathrm{B}}$ per Yb. Moreover, a crystalline approximant to this quasicrystal is an IV compound. We propose a heterogeneous IV model for the quasicrystal, whereas the crystalline approximant is most likely a homogeneous IV system. At temperatures below $\ensuremath{\sim}10$ K, specific heat and magnetization measurements reveal non-Fermi-liquid behavior in both the quasicrystal and its crystalline approximant without either doping, pressure, or field tuning.

Abstract: In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Backlund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Backlund transformations, which produce N-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions.

Abstract: As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

Abstract: Transient quantum hyperdiffusion, namely, faster-than-ballistic wave packet spreading for a certain time scale, is found to be a typical feature in tight-binding lattices if a sublattice with on-site potential is embedded in a uniform lattice without on-site potential. The strength of the sublattice on-site potential, which can be periodic, disordered, or quasiperiodic, must be below certain threshold values for quantum hyperdiffusion to occur. This is explained by an energy band mismatch between the sublattice and the rest uniform lattice and by the structure of the underlying eigenstates. Cases with a quasiperiodic sublattice can yield remarkable hyperdiffusion exponents that are beyond three. A phenomenological explanation of hyperdiffusion exponents is also discussed.

Abstract: We study the time evolution of wave packets in one-dimensional quasiperiodic lattices which localize linear waves. Nonlinearity (related to two- body interactions) has a destructive effect on localization, as observed recently for interacting atomic condensates (Lucioni et al 2011 Phys. Rev. Lett. 106 230403). We extend the analysis of the characteristics of the subdiffusive dynamics to large temporal and spatial scales. Our results for the second moment m2 consistently reveal an asymptotic m2 t 1/3 and an intermediate m2 t 1/2 law. At variance with purely random systems (Laptyeva et al 2010 Europhys. Lett. 91 30001), the fractal gap structure of the linear wave spectrum strongly favours intermediate self-trapping events. Our findings give a new dimension to the theory of wave packet spreading in localizing environments.

Abstract: Nanophotonic structures with irregular symmetry, such as quasiperiodic plasmonic crystals, have gained an increasing amount of attention, in particular as potential candidates to enhance the absorption of solar cells in an angular insensitive fashion. To examine the photonic bandstructure of such systems that determines their optical properties, it is necessary to measure and model normal and oblique light interaction with plasmonic crystals. We determine the different propagation vectors and consider the interaction of all possible waveguide modes and particle plasmons in a 2D metallic photonic quasicrystal, in conjunction with the dispersion relations of a slab waveguide. Using a Fano model, we calculate the optical properties for normal and inclined light incidence. Comparing measurements of a quasiperiodic lattice to the modelled spectra for angle of incidence variation in both azimuthal and polar direction of the sample gives excellent agreement and confirms the predictive power of our model.

Abstract: In the present work, we study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a time dependent magnetic field using the Landau-Lifshitz-Gilbert equation. In particular, we study the case when the magnetic field consists in two terms. One is perpendicular to the anisotropy direction and has quasiperiodic time dependence, while the other term is constant and parallel to the anisotropy direction. We numerically characterize the dynamical behavior of the system by monitoring the Lyapunov exponents, and by calculating Poincare sections and Fourier spectra. In addition, we calculate analytically the corresponding Melnikov function which gives us the bifurcations of the homoclinic orbits. We find a rather complicated landscape of sometimes closely intermingled chaotic and non-chaotic areas in parameters space. Finally, we show that the system exhibits strange nonchaotic attractors.

Abstract: In this paper, we first obtain a formula of averaged Lyapunov exponents for ergodic Szegő cocycles via the Herman–Avila–Bochi formula. Then using acceleration, we construct a class of analytic quasiperiodic Szegő cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in Young (1997 Ergod. Theory Dyn. Syst. 25 483–504) allows us to estimate the Lebesgue measure of support of the measure associated with certain class of C1 quasiperiodic 2-sided Verblunsky coefficients. Using the same method, we also recover the Sorets and Spencer (1991 Commun. Math. Phys. 142 543–66) results for Schrodinger cocycles with nonconstant real analytic potentials and obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno frequency and for certain C1 potentials.

Abstract: We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic $${GL(d,\mathbb R)}$$ cocycles which are close to constants It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes

Abstract: This paper presents a novel bounded four-dimensional (4D) chaotic system which can display hyperchaos, chaos, quasiperiodic and periodic behaviors, and may have a unique equilibrium, three equilibria and five equilibria for the different system parameters. Numerical simulation shows that the chaotic attractors of the new system exhibit very strange shapes which are distinctly different from those of the existing chaotic attractors. In addition, we investigate the ultimate bound and positively invariant set for the new system based on the Lyapunov function method, and obtain a hyperelliptic estimate of it for the system with certain parameters.

Abstract: We study the spatio-temporal evolution of wave packets in one-dimensional quasiperiodic lattices which localize linear waves. Nonlinearity (related to two-body interactions) has destructive effect on localization, as recently observed for interacting atomic condensates [Phys. Rev. Lett. 106, 230403 (2011)]. We extend the analysis of the characteristics of the subdiffusive dynamics to large temporal and spatial scales. Our results for the second moment $m_2$ consistently reveal an asymptotic $m_2 \sim t^{1/3}$ and intermediate $m_2 \sim t^{1/2}$ laws. At variance to purely random systems [Europhys. Lett. 91, 30001 (2010)] the fractal gap structure of the linear wave spectrum strongly favors intermediate self-trapping events. Our findings give a new dimension to the theory of wave packet spreading in localizing environments.

Abstract: We consider an integer lattice quasiperiodic Schrodinger operator. The underlying dynamics is either the skew-shift or the multi-frequency shift by a Diophantine frequency. We assume that the potential function belongs to a Gevrey class on the multi-dimensional torus. Moreover, we assume that the potential function satisfies a generic transversality condition, which we show to imply a Lojasiewicz type inequality for smooth functions of several variables. Under these assumptions and for large coupling constant, we prove that the associated Lyapunov exponent is positive for all energies, and continuous as a function of energy, with a certain modulus of continuity. Moreover, in the large coupling constant regime and for an asymptotically large frequency - phase set, we prove that the operator satisfies Anderson localization.

Abstract: In this work, we report the experimental results and theoretical analysis of strong localization of resonance transmission modes generated by hybrid periodic/quasiperiodic heterostructures (HHs) based on porous silicon. The HHs are formed by stacking a quasiperiodic Fibonacci (FN) substructure between two distributed Bragg reflectors (DBRs). FN substructure defines the number of strong localized modes that can be tunable at any given wavelength and be unfolded when a partial periodicity condition is imposed. These structures show interesting properties for biomaterials research, biosensor applications and basic studies of adsorption of organic molecules. We also demonstrate the sensitivity of HHs to material infiltration.

Abstract: A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

Abstract: The scattering of a time-harmonic plane elastic wave by a two-dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasiperiodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies. Copyright © 2010 John Wiley & Sons, Ltd.

Abstract: Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincare map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.

Abstract: We study discrete quasiperiodic Schr\"odinger operators on $\ell^2(\zee)$ with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.

Abstract: This paper is concerned with the inverse electromagnetic scattering by a 2D (impenetrable or penetrable) smooth periodic curve. Precisely, we establish global uniqueness results on the inverse problem of determining the grating profile from the scattered fields corresponding to a countably infinite number of quasiperiodic incident waves. For the case of an impenetrable and partially coated perfectly reflecting grating, we prove that the grating profile and its physical property can be uniquely determined from the scattered field measured above the periodic structure. For the case of a penetrable grating, we show that the periodic interface can be uniquely recovered by the scattered field measured only above the interface. A key ingredient in our proofs is a novel mixed reciprocity relation that is derived in this paper for the periodic structures and seems to be new. Copyright © 2012 John Wiley & Sons, Ltd.

Abstract: We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincare section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.

Abstract: Consider the scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface in R 3. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while the material below is perfectly conducting. This paper is concerned with the existence of quasiperiodic solutions for any frequency. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves. The only assumption is that the grating profile is a Lipschitz biperiodic surface. Note that the solvability result of the present paper covers the resonance case where Rayleigh frequencies are allowed. Finally, non-uniqueness examples are presented in the resonance case and in the case of TE or TM polarization for classical gratings.

Abstract: In this paper we consider one-dimensional classical and quantum spin-1=2 quasiperiodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field, by employing the techniques that were developed in our previous work In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibonacci substitution (thus proving absence of phase transitions of any order at finite temperature) We also investigate the distribution of Lee-Yang zeros of the partition function in the complex magnetic field regime, and prove its Cantor set structure (together with some additional qualitative properties), thus providing a rigorous justification for the observations in some previous works In both, quantum and classical models, we concentrate on the ferromagnetic class