Abstract: The influence of linear time-delayed feedback on vibrational resonance is investigated in underdamped and overdamped Duffing oscillators with double-well and single-well potentials driven by both low frequency and high frequency periodic forces. This task is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay time at which resonance occurs are in very good agreement with the numerical simulation. A major consequence of time-delayed feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the time-delayed parameter. An appropriate time delay is shown to induce a resonance in an overdamped single-well system which is otherwise not possible. For a range of values of the time-delayed parameters, the response amplitude is found to be larger than in delay-time feedback-free systems.

Abstract: The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Karman equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincare maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.

Abstract: We propose a new holographic model of Josephson junctions (and networks thereof) based on designer multi-gravity, namely multi-(super)gravity theories on products of distinct asymptotically AdS spacetimes coupled by mixed boundary conditions. We present a simple model of a Josephson junction (JJ) that reproduces trivially the well-known current-phase sine relation of JJs. In one-dimensional chains of holographic superconductors we find that the Cooper-pair condensates are described by a discretized Schrodinger-type equation. Such non-integrable equations, which have been studied extensively in the past in condensed matter and optics applications, are known to exhibit complex behavior that includes periodic and quasiperiodic solutions, chaotic dynamics, soliton and kink solutions. In our setup these solutions translate to holographic configurations of strongly-coupled superconductors in networks with weak site-to-site interactions that exhibit interesting patterns of modulated superconductivity. In a continuum limit our equations reduce to generalizations of the Gross-Pitaevskii equation. We comment on the many possible extensions and applications of this new approach.

Abstract: Whereas periodic gratings enable us to couple light into a surface plasmon polariton only at a specific angle and wavelength, we show here that quasiperiodic gratings enable the coupling of light at multiple wavelengths and angles. The quasiperiodic grating can be designed in a systematic manner using the dual-grid method, thereby enabling us to control the coupling strength and grating dimensions. We verified the method experimentally by efficiently coupling light into a surface plasmon from several different illumination angles using a single quasiperiodic grating.

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 non-zero 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 for the computation of GALIs the components of deviation vectors orthogonal to the direction of motion, the indices of stable periodic orbits behave for flows as they do for maps.

Abstract: We report on the experimental evidence of Fano-type guided resonances (GRs) in aperiodically-ordered photonic quasicrystal slabs. With specific reference to the Ammann-Beenker (8-fold symmetric, quasiperiodic) octagonal tiling geometry, we present our experimental results on silicon-on-insulator devices operating at near-infrared wavelengths, and compare them with the full-wave numerical predictions based on periodic approximants. Our results indicate that spatial periodicity is not strictly required for the GR excitation, and may be effectively surrogated by weaker forms of long-range aperiodic order which intrinsically provide extra degrees of freedom (e.g., higher-order rotational symmetries, richer defect states and phase-matching conditions, etc.) to be exploited in the design and performance optimization of nanostructured dielectric slabs operating in the out-of-plane configuration. The essential spectral features may be qualitatively understood in terms of phase-matching conditions derived from approximate homogenized models, and turn out to be effectively captured by full-wave modeling based on suitably-sized periodic approximants.

Abstract: Stationary and oscillatory bound states, or complexes, of the damped-driven solitons are numerically path-followed in the parameter space. We compile a chart of the two-soliton attractors, complementing the one-soliton attractor chart.

Abstract: We demonstrate a position-sensing technique that relies on the inherent sensitivity of chaos, where we illuminate a subwavelength object with a complex structured radio-frequency field generated using wave chaos and nonlinear feedback. We operate the system in a quasiperiodic state and analyze changes in the frequency content of the scalar voltage signal in the feedback loop. This allows us to extract the object's position with a one-dimensional resolution of $\ensuremath{\sim}\ensuremath{\lambda}/10\text{ }000$ and a two-dimensional resolution of $\ensuremath{\sim}\ensuremath{\lambda}/300$, where $\ensuremath{\lambda}$ is the shortest wavelength of the illuminating source.

Abstract: One of the most important issues in disordered systems is the interplay of the disorder and repulsive interactions. Several recent experimental advances on this topic have been made with ultracold atoms, in particular the observation of Anderson localization and the realization of the disordered Bose–Hubbard model. There are, however, still questions as to how to differentiate the complex insulating phases resulting from this interplay, and how to measure the size of the superfluid fragments that these phases entail. It has been suggested that the correlation function of such a system can give new insights, but so far very little experimental investigation has been performed. Here, we show the first experimental analysis of the correlation function for a weakly interacting, bosonic system in a quasiperiodic lattice. We observe an increase in the correlation length as well as a change in the shape of the correlation function in the delocalization crossover from Anderson glass to coherent, extended state. In between, the experiment indicates the formation of progressively larger coherent fragments, consistent with a fragmented BEC, or Bose glass.

Abstract: The role of magnetic domains and domain walls in exchange bias has stimulated much contemporary deliberation. Here we present compelling evidence obtained with small-angle scattering of unpolarized- and polarized-neutron beams that magnetization reversal occurs via formation of 10--100s nm-sized magnetic domains in an exchange-biased DyFe${}_{2}$/YFe${}_{2}$ superlattice. The reversal mechanism is observed to involve rotation of magnetization in and out of the sample plane. Remarkably, the domains are arranged in a quasiperiodic manner in the plane of the sample. The length scale of domain formation is similar to that of structural defects at the seed-layer-superlattice interface.

Abstract: A Lagrangian formalism is developed for a general nondissipative quasiperiodic nonlinear wave with trapped particles in collisionless plasma. The adiabatic time-averaged Lagrangian density $\mcc{L}$ is expressed in terms of the single-particle oscillation-center Hamiltonians; once those are found, the complete set of geometrical-optics equations is derived without referring to the Maxwell-Vlasov system. The number of trapped particles is assumed fixed; in particular, those may reside close to the bottom of the wave trapping potential, so they never become untrapped. Then their contributions to the wave momentum and the energy flux depend mainly on the trapped-particle density, as an independent parameter, and the phase velocity rather than on the wave amplitude $a$ explicitly; hence, $\mcc{L}$ acquires $a$-independent terms. Also, the wave action is generally not conserved, because it can be exchanged with resonant oscillations of the trapped-particle density. The corresponding modification of the wave envelope equation is found explicitly, and the new action flow velocity is derived. Applications of these results are left to the other two papers of the series, where specific problems are addressed pertaining to properties and dynamics of waves with trapped particles.

Abstract: A group-theoretical approach for studying localized periodic and quasiperiodic vibrations in two- and three-dimensional lattice dynamical models is developed. This approach is demonstrated for the scalar models on the plane square lattice. The symmetry-determined invariant manifolds admitting existence of localized vibrations are found, and some types of discrete breathers are constructed on these manifolds. A general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed. This method allows one to decompose the corresponding system of linear differential equations with time-dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system.

Abstract: In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N-degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e. periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1, 2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature. We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies Eq of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, …. These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALIk, k = 2, …, 2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.

Abstract: Simulation models for the optical properties of 2D quasiperiodic plasmonic structures often fail due to their lack of periodicity. Therefore, it is necessary to find an appropriate model to describe the optical properties of such structures. In this paper we present a model which is able to describe the optical spectra of 2D metallic photonic quasicrystals on top of a waveguide. We take the 2D Fourier transform of the structure and consider all possible waveguide modes in the specific energy range. By utilizing the dispersion relations, the optical spectra can be calculated. The presented model is verified by measurements of a quasiperiodic lattice as well as a rectangular lattice as reference. We find distinct differences in the behavior of quasicrystalline vs rectangular lattices, in particular when investigating rotated and elongated plasmonic particles.

Abstract: I Conformally Invariant Field Theories, Integrability, Quantum Groups- Z/NZ Conformal Field Theories- Affine Characters and Modular Transformations- Conformal Field Theory on a Riemann Surface- Modular Invariance of Field Theories and String Compactifications- Yang-Baxter Algebras and Quantum Groups- Representations of Uq sl(2) for q a Root of Unity- II Quasicrystals and Related Geometrical Structures- Some Quasiperiodic Tilings as Modulated Lattices- Types of Order and Diffraction Spectra for Tilings of the Line- A Topological Constraint on the Atomic Structure of Quasicrystals- From Approximants to Quasicrystals: A Non-standard Approach- The Topological Structure of Grain Boundaries- Calculation of 6D Atomic Surfaces from a Given Approximant Crystalline Structure Using Approximate Icosahedral Periodic Tilings- III Spectral Problems, Automata and Substitutions- Spectral Properties of Schrodinger's Operator with a Thue-Morse Potential- On the Non-commutative Torus of Real Dimension Two- Topological Invariants Associated with Quasi-Periodic Quantum Hamiltonians- Spectra of Some Almost Periodic Operators- The Quantum Hall Effect and the Schrodinger Equation with Competing Periods- Finite Automata in 1-D and 2-D Physics- Summation Formulae for Substitutions on a Finite Alphabet- The Inhomogeneous Ising Chain and Paperfolding- IV Dynamical and Stochastic Systems- A Nonlinear Evolution with Travelling Waves- Iterating Random Maps and Applications- Hannay Angles and Classical Perturbation Theory- Nekhoroshev Stability Estimates for Symplectic Maps and Physical Applications- p-adic Dynamical Systems- V Further Arithmetical Problems, and Their Relationship to Physics- Dirichlet Series Associated with a Polynomial- Some Remarks on Random Number Generators- Bounds for Non-blocking Switch Networks- The Ising Model and the Diophantine Moment Problem- Statistical Theory of Numbers- Algebraic Number Theory and Hamiltonian Chaos- Semiclassical Properties of the Cat Maps- Index of Contributors

Abstract: A quasiperiodic intermediate-valence system was prepared by applying pressure to an icosahedral Cd-Mg-Yb quasicrystal. X-ray absorption spectroscopy near the Yb ${L}_{3}$ edge demonstrates that the Yb valence increases continuously upon compression from the divalent state ($4{f}^{14}$) at ambient pressure and reaches a value of 2.71 at 57.6 GPa, which is close to the trivalent state ($4{f}^{13}$). By following the trend of Yb-based intermediate-valence crystalline compounds, this large valence increase suggests the change of $4f$ character in the Cd-Mg-Yb quasicrystal from an itinerant state with weak electron correlations to an intermediate region between an itinerant and a localized state with strong correlations. The valence increases sensitively with pressure below $\ensuremath{\sim}$30 GPa; however, the increase is significantly suppressed above $\ensuremath{\sim}$30 GPa. The rate of valence increase with respect to pressure in the lower-pressure region is twice larger than that of a Cd-Yb quasicrystal in our previous study [Phys. Rev. B 81, 220202(R) (2010)]. This is mainly explained by the smaller bulk modulus of the Cd-Mg-Yb quasicrystal compared to the Cd-Yb system. The suppression of valence increase with respect to pressure in the higher-pressure region is most likely due to an increase of conduction-$4f$ electron hybridization that counteracts $4f$ localization and reduces the Yb valence increase.

Abstract: We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.

Abstract: We imaged Abrikosov vortex patterns in thin Nb films with random, periodic (triangular), and quasiperiodic (Penrose) arrays of antidots. Vortex positions were visualized by Bitter decoration for a range of applied fields $B$, antidot radii $r$, and densities ${n}_{p}$ after field-cooling through the transition temperature ${T}_{c}$ to a base temperature $T\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The observed vortex patterns correspond to snapshots of vortex positions at the time of decoration. The effectiveness of antidots as artificial pinning sites for vortices is found to be sensitive to several factors: array geometry, antidot size and density, and applied field. Overall, the triangular lattice provides the most effective pinning landscape, with antidots trapping the highest proportion of vortices, but for a wide range of parameters the Penrose lattice is equally effective. For a quantitative analysis, we determined the occupation number $n$ (average number of vortices trapped per antidot) from each image. This revealed a significantly more complicated dependence of antidot occupation on applied field and/or antidot density than that predicted by simple models considering pinning by an isolated antidot. In particular, upon increasing the antidot density ${n}_{p}$, we find a marked increase in $n$ for triangular arrays, which we attribute to the additional repulsion from interstitial vortices, pushing more vortices into antidots with decreasing antidot separation. This effect is also present but less pronounced for Penrose arrays, which can be explained by the variation of antidot spacing inherent to the Penrose geometry and accordingly more options for accommodating interstitial vortices.

Abstract: Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $$ \mathcal{N} = 1 $$ supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an N-periodic wave solution with arbitrary parameters for N ≥ 2. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.

Abstract: [1] We develop a theory of formation of a fine structure in the dynamic spectra of the Jovian decametric radio emission. Main attention is paid to the formation of narrowband (NB) emission and quasiperiodic trains of short (S) bursts. Our model is based on the effects of occurrence of the amplitude-frequency modulation and extension of the frequency spectrum of a signal during propagation of radiation in a medium with time-varied parameters. It is shown that nonstationary disturbances of the planetary magnetic field and strong frequency dispersion of the plasma at frequencies close to the cutoff frequency of the extraordinary wave in the Jovian ionosphere play a crucial role in the formation of NB emission and quasiperiodic trains of S bursts. As a result of the numerical experiments, it was concluded that the amplitude-frequency characteristics of an initially continuous signal can drastically vary as a functions of the form of the magnetic field disturbance in the Jovian ionosphere. Structures similar to those observed in the real experiments, ranging from NB emission and quasiperiodic trains of S bursts to more complex structures, arise in the dynamic spectrum. Time variation in the conditions of generation and propagation of decametric radiation in the Jovian ionosphere is reflected in the dynamic spectrum as a time variation in the fine structure of the radiation. For example, a structure of the NB emission type is replaced by a quasiperiodic train of S bursts and vice versa.

Abstract: Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.

Abstract: A comparison of transmission spectra from periodic, quasiperiodic, and randomly spaced slit arrays in thick gold films reveals resonant plasmonic excitations that arise solely due to the long-range order of the quasiperiodic structures. Specifically, first-order plasmonic resonances at the air-gold interface of the quasiperiodic arrays are identified at a broader range of wavelengths than those observed from periodic structures with the same average slit distance. Thus, a quasiperiodic plasmonic coupler that couples both visible and near-infrared light to surface plasmon polaritons is designed and demonstrated.

Abstract: Characterization for nonpremixed biodiesel/air jet flames instability is investigated by the 0-1 test for chaos and recurrence plots. Test conditions involve biodiesel from Jatropha curcas. L-fueled flames have inlet oil pressure of 0.2–0.6 MPa, fuel flow rates (Q1) of 15–30 kg/h, and combustion air flow rate (Q2) of 150–750 m3/h. This method is based on image analysis and nonlinear dynamics. Structures of flame are analyzed using an image analysis technique to extract position series which are representative of the relative change in temperature of combustion chamber. Compared with the method of maximum Lyapunov exponent, the 0-1 test succeeds in detecting the presence of regular and chaotic components in flame position series. Periodic and quasiperiodic characteristics are obtained by the Poincare sections. A common characteristic of regular nonpremixed flame tip position series is detected by recurrence plots. Experimental results show that these flame oscillations follow a route to chaos via periodic and quasiperiodic states.

Abstract: The description of almost periodic or quasiperiodic structures has a long tradition in mathematical physics, in particular since the discovery of quasicrystals in the early 80's. Frequently, the modelling of such structures leads to different types of dynamical systems which include, depending on the concept of quasiperiodicity being considered, skew products over quasiperiodic or almost-periodic base flows, mathematical quasicrystals or maps of the real line with almost-periodic displacement. An important problem in this context is to know whether the considered system is semiconjugate to a rigid translation. We solve this question in a general setting that includes all the above-mentioned examples and also allows to treat scalar differential equations that are almost-periodic both in space and time. To that end, we study a certain class of flows that preserve a one-dimensional foliation and show that a semiconjugacy to a minimal translation flow exists if and only if a boundedness condition, concerning the distance of orbits of the flow to those of the translation, holds.

Abstract: In 1988 we discovered generalized grid--projection method. Since then the method proved to be useful for description of symmetries of quasicrystals also for analysis of interacting spins.

Abstract: A group-theoretical approach for studying localized periodic and quasiperiodic vibrations in 2D and 3D lattice dynamical models is developed. This approach is demonstrated for the scalar models on the plane square lattice. The symmetry-determined invariant manifolds admitting existence of localized vibrations are found and some types of discrete breathers are constructed on these manifolds. A general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed. This method allows one to decompose the corresponding system of linear differential equations with time-dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system.

Abstract: We consider the trace map associated with the silver ratio Schrodinger operator as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently large. As a consequence, for this values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of this operator all coincide and are smooth functions of the coupling constant.