Abstract: We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional $p$-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space.

Abstract: We show that the recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors(SNAs) and periodic/quasiperiodic/chaotic attractors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number which avoids resonance with the quasiperiodic force. We further characterize the transition from quasiperiodic to chaotic motion via SNAs interms of the 0-1 test. We demonstrate that the test helps to detect different dynamical transitions to SNAs from quasiperiodic attractor or the transitions from SNAs to chaos. We illustrate the performance of the 0-1 test in detecting transitions to SNAs in quasiperiodically forced logistic map, cubic map, and Duffing oscillator.

Abstract: We study the regularity of the Lyapunov exponent for quasiperiodic cocycles (Tω,A) where Tω is an irrational rotation x→x+2πω on S1 and A∈Cl(S1,SL(2,R)), 0≤l≤∞. For any fixed l=0,1,2,…,∞ and any fixed ω of bounded type, we construct Dl∈Cl(S1,SL(2,R)) such that the Lyapunov exponent is not continuous at Dl in Cl-topology. We also construct such examples in a smaller Schrodinger class.

Abstract: We show that the recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors (SNAs) and periodic/quasiperiodic/chaotic attractors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number which avoids resonance with the quasiperiodic force. We further characterize the transition from quasiperiodic to chaotic motion via SNAs in terms of the 0-1 test. We demonstrate that the test helps to detect different dynamical transitions to SNAs from quasiperiodic attractor or the transitions from SNAs to chaos. We illustrate the performance of the 0-1 test in detecting transitions to SNAs in quasiperiodically forced logistic map, cubic map, and Duffing oscillator.

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. Finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

Abstract: We discuss a numerical method for the calculation of the spectrum of Lyapunov exponents for spatially extended systems described by coupled Poisson and continuity equations. This approach was applied to the model of collective charge transport in semiconductor superlattices operating in the miniband transport regime. The method is in very good agreement with analytical results obtained for the steady state. As an illustrative example, we consider the collective electron dynamics in the superlattice subjected to an ac voltage and a tilted magnetic field, and conclusively show that, depending on the field parameters, the dynamics can exhibit periodic, quasiperiodic, or chaotic behavior.

Abstract: We present numerical solutions of the flow in precessing spheres and spherical shells with small (${r}_{i}/{r}_{o}=0.1$) inner cores and either stress-free or no-slip inner boundary conditions. For each of these three cases we consider the sequence of bifurcations as the Reynolds number $\text{Re}={r}_{o}^{2}\ensuremath{\Omega}/\ensuremath{ u}$ is increased up to $\ensuremath{\sim}$1280, focusing particular attention on bifurcations that break the antipodal symmetry $\mathbf{U}(\ensuremath{-}\mathbf{r})=\ensuremath{-}\mathbf{U}(\mathbf{r})$. All three cases have steady and time-periodic symmetric solutions at smaller Re, and quasiperiodic asymmetric solutions at larger Re, but the details of the transitions differ, and include also periodic asymmetric and quasiperiodic symmetric solutions in some of the cases.

Abstract: Quasiperiodic oscillations and shape-transformations of higher-order bright solitons in nonlinear nonlocal media have been frequently observed numerically in recent years, however, the origin of these phenomena was never completely elucidated. In this paper, we perform a linear stability analysis of these higher-order solitons by solving the Bogoliubov–de Gennes equations. This enables us to understand the emergence of a new oscillatory state as a growing unstable mode of a higher-order soliton. Using dynamically important states as a basis, we provide low-dimensional visualizations of the dynamics and identify quasiperiodic and homoclinic orbits, linking the latter to shape-transformations.

Abstract: Natural convection of air between two infinite vertical differentially heated plates is studied analytically in two dimensions (2D) and numerically in two and three dimensions (3D) for Rayleigh numbers $\mathrm{Ra}$ up to 3 times the critical value ${\mathrm{Ra}}_{c}=5708$. The first instability is a supercritical circle pitchfork bifurcation leading to steady 2D corotating rolls. A Ginzburg-Landau equation is derived analytically for the flow around this first bifurcation and compared with results from direct numerical simulation (DNS). In two dimensions, DNS shows that the rolls become unstable via a Hopf bifurcation. As $\mathrm{Ra}$ is further increased, the flow becomes quasiperiodic, and then temporally chaotic for a limited range of Rayleigh numbers, beyond which the flow returns to a steady state through a spatial modulation instability. In three dimensions, the rolls instead undergo another pitchfork bifurcation to 3D structures, which consist of transverse rolls connected by counter-rotating vorticity braids. The flow then becomes time dependent through a Hopf bifurcation, as exchanges of energy occur between the rolls and the braids. Chaotic behavior subsequently occurs through two competing mechanisms: a sequence of period-doubling bifurcations leading to intermittency or a spatial pattern modulation reminiscent of the Eckhaus instability.

Abstract: We study the properties of wavefunctions and the wavepacket dynamics in quasiperiodic tight-binding models in one, two, and three dimensions. The atoms in the one-dimensional quasiperiodic chains are coupled by weak and strong bonds aligned according to the Fibonacci sequence. The associated d-dimensional quasiperiodic tilings are constructed from the direct product of d such chains, which yields either the hypercubic tiling or the labyrinth tiling. This approach allows us to consider fairly large systems numerically. We show that the wavefunctions of the system are multifractal and that their properties can be related to the structure of the system in the regime of strong quasiperiodic modulation by a renormalization group (RG) approach. We also study the dynamics of wavepackets to get information about the electronic transport properties. In particular, we investigate the scaling behaviour of the return probability of the wavepacket with time. Applying again the RG approach we show that in the regime of strong quasiperiodic modulation the return probability is governed by the underlying quasiperiodic structure. Further, we also discuss lower bounds for the scaling exponent of the width of the wavepacket and propose a modified lower bound for the absolute continuous regime.

Abstract: We first present some sufficient conditions for the existence of a uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences. Then we study the existence of uniform exponential attractors for the dissipative non-autonomous Klein–Gordon–Schrodinger lattice system and the Zakharov lattice system driven by quasi-periodic external forces in the space of infinite sequences.

Abstract: This paper addresses the behavior of large systems of heterogeneous, globally coupled oscillators each of which is described by the generic Landau-Stuart equation, which incorporates both phase and amplitude dynamics of individual oscillators. One goal of our paper is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and of the effect of a homogeneous nonlinear frequency shift. Both of these effects are of potential relevance to recently reported experiments. Our second goal is to gain further understanding of the macroscopic system dynamics at large coupling strength, and its dependence on the nonlinear frequency shift parameter. It is proven that at large coupling strength, if the nonlinear frequency shift parameter is below a certain value, then there is a unique attractor for which the oscillators all clump at a single amplitude and uniformly rotating phase (we call this a single-cluster “locked state”). Using a combination of analytical and numerical methods, we show that at higher values of the nonlinear frequency shift parameter, the single-cluster locked state attractor continues to exist, but other types of coexisting attractors emerge. These include two-cluster locked states, periodic orbits, chaotic orbits, and quasiperiodic orbits.

Abstract: Understanding the routes to chaos occurring in atmospheric-pressure dielectric barrier discharge systems by changing controlling parameters is very important to predict and control the dynamical behaviors. In this paper, a route of a quasiperiodic torus to chaos via the strange nonchaotic attractor is observed in an atmospheric-pressure dielectric barrier discharge driven by triangle-wave voltage. By increasing the driving frequency, the discharge system first bifurcates to a quasiperiodic torus from a stable single periodic state, and then torus and phase-locking periodic state appear and disappear alternately. In the meantime, the torus becomes increasingly wrinkling and stretching, and gradually approaches a fractal structure with the nonpositive largest Lyapunov exponent, i.e., a strange nonchaotic attractor. After that, the discharge system enters into chaotic state. If the driving frequency is further increased, another well known route of period-doubling bifurcation to chaos is also observed.

Abstract: By investigating the magnetism of spins on a quasiperiodic lattice, we present an experimental study of static and dynamic magnetic properties, specific heat, and magnetic entropy of the Gd${}_{3}$Au${}_{13}$Sn${}_{4}$ quasicrystalline approximant. The magnetic sublattice of Gd${}_{3}$Au${}_{13}$Sn${}_{4}$ is a periodic arrangement of nonoverlapping spin clusters of almost perfect icosahedral symmetry, where gadolinium localized $f$ magnetic moments are distributed on equilateral triangles. The absence of disorder on the magnetic sublattice and the antiferromagnetic (AFM) interactions between the nearest-neighbor spins distributed on triangles result in geometrical frustration of spin-spin interactions. Thus, the Gd${}_{3}$Au${}_{13}$Sn${}_{4}$ phase can be viewed as a prototype site-ordered, geometrically frustrated spin system on icosahedral clusters. The zero-field-cooled and field-cooled magnetic susceptibilities, the alternating current susceptibility, the thermoremanent magnetization, the memory effect, the magnetic specific heat, and the magnetic entropy all show that the spin system undergoes at low temperatures a transition to a nonergodic state at the spin freezing temperature ${T}_{f}$ \ensuremath{\approx} 2.8 K. Below this, the ergodicity is broken on the experimental timescale, because the thermally activated correlation times for the spin reorientations become macroscopically long. The magnetic state achieved at low temperatures by continuous cooling in low magnetic fields is likely a superposition of (1) metastable states with randomly frozen spins that have no long-range order yet undergo gradual spin-freezing dynamics and (2) an AFM-like magnetically ordered state with critical slowing dynamics. The magnetic properties of the site-ordered, geometrically frustrated Gd${}_{3}$Au${}_{13}$Sn${}_{4}$ system are discussed in comparison to site-disordered spin glasses that contain both randomness and frustration.

Abstract: We introduce a construction to "periodize" a quasiperiodic lattice of obstacles, i.e., embed it into a unit cell in a higher-dimensional space, reversing the projection method used to form quasilattices. This gives an algorithm for simulating dynamics, as well as a natural notion of uniform distribution, in quasiperiodic structures. It also shows the generic existence of channels, where particles travel without colliding, up to a critical obstacle radius, which we calculate for a Penrose tiling. As an application, we find superdiffusion in the presence of channels, and a subdiffusive regime when obstacles overlap.

Abstract: The work is devoted to analytic and numeric investigation of dynamical behavior in a system of two Van der Pol (VdP) oscillators coupled by a non-dispersive elastic rod. The model is rigorously reduced to a system of nonlinear neutral differential delay equations. For the case of relatively small coupling and moderate delay, an approximate analytic investigation can be accomplished by means of an averaging procedure. The region of synchronization in the space of parameters is established and characteristic bifurcations are revealed. A numeric study confirms the validity of the analytic approach in the synchronization region. Beyond this region, the averaging approach is no more valid. A multitude of quasiperiodic and chaotic-like orbits has been revealed. Especially interesting behavior occurs in the case of relatively large delays and corresponds to sequential quenching and excitation of the VdP oscillators. This regime is also explored analytically, by means of a large-delay approximation, which reduces the system to a perturbed discrete map.

Abstract: This paper is concerned with the variational approach in weighted Sobolev spaces to time-harmonic elastic wave scattering by one-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results at arbitrary frequency for both elastic plane wave and point source (spherical) wave incidence in the two-dimensional case. In particular, our approach covers the elastic scattering from periodic structures (diffraction gratings), and we prove quasiperiodicity of the scattered field whenever the incident field is quasiperiodic. Moreover, the diffraction grating problem is also uniquely solvable in the presented weighted Sobolev spaces for a broad class of non-quasiperiodic incident waves.

Abstract: We investigate the nonlinear dynamics of long-wave Marangoni convection in a 2D binary-liquid layer heated from below. Free surface deformations and the Soret effect are taken into account. We employ the set of evolution equations derived in earlier work in the case of small Galileo and Lewis numbers and solve it numerically with periodic boundary conditions. We validate our numerical solution by comparison between the results obtained via two different numerical methods, as well as by comparison with the prior analytical results. We study the transitions between the nonlinear regimes emerging at finite supercriticality values and find a rich variety of patterns. In a sufficiently large computational domain, we observe multistability of waves chaotic in time and spatially replicated periodic and quasiperiodic solutions. For sufficiently high values of the Marangoni number, we also observe a breakdown of model equations.

Abstract: The quantum mechanical transmission probability is calculated for one-dimensional finite lattices with three types of potentials: periodic, quasiperiodic, and random. When the number of lattice sites included in the computation is systematically increased, distinct features in the transmission probability vs. energy diagrams are observed for each case. The periodic lattice gives rise to allowed and forbidden transmission regions that correspond to the energy band structure of the infinitely periodic potential. In contrast, the transmission probability diagrams for both quasiperiodic and random lattices show the absence of well-defined band structures and the appearance of wave localization effects. Using the average transmissivity concept, we show the emergence of exponential (Anderson) and power-law bounded localization for the random and quasiperiodic lattices, respectively.

Abstract: We propose a means to realize two-dimensional quasiperiodic structures by trapping atoms in an optical potential. The structures have eight-fold symmetry and are closely related to the well-known quasiperiodic octagonal (Ammann-Beenker) tiling. We describe the geometrical properties of the structures obtained by tuning parameters of the system. We discuss some features of the corresponding tight-binding models, and experiments to probe quantum properties of this optical quasicrystal.

Abstract: In this paper, an omnidirectional photonic band gap (OBG) of one-dimensional (1D) superconductor–dielectric photonic crystals (SDPCs) with a modified ternary Fibonacci quasiperiodic structure which originates from Bragg gap is theoretically investigated by the transfer matrix method (TMM) in detail. The SDPCs are composed of superconductor and two kinds of homogeneous, isotropic dielectric. Such OBG is independent of the incidence angle and the polarization of electromagnetic wave (EM wave). From the numerical results, the OBG can be notably enlarged by tuning the thickness of superconductor and dielectric layers but cease to change with increasing the Fibonacci order. The OBG also can be manipulated by the ambient temperature of system. Especially, the ambient temperature of system is close to the critical temperature. However, the damping coefficient of superconductor has no effects on the OBG. The gap/midgap ratio of OBG also is studied by those parameters. It is shown that 1D SDPCs with a modified ternary Fibonacci quasiperiodic structure have a superior feature in the enhancement of OBG compared with the conventional 1D ternary and conventional ternary Fibonacci quasiperiodic SDPCs.

Abstract: Textures (i.e., smooth space nonuniform distributions of the order parameter) in biaxial nematics turned out to be much more complex and interesting than expected. Scanning the literature we find only a very few publications on this topic. Thus, the immediate motivation of the present paper is to develop a systematic procedure to study, classify, and visualize possible textures in biaxial nematics. Based on the elastic energy of a biaxial nematic (written in the most simple form that involves the least number of phenomenological parameters) we derive and solve numerically the Lagrange equations of the first kind. It allows one to visualize the solutions and offers a deep insight into their geometrical and topological features. Performing Fourier analysis we find some particular textures possessing two or more characteristic space periods (we term such solutions quasiperiodic ones because the periods are not necessarily commensurate). The problem is not only of intellectual interest but also of relevance to optical characteristics of the liquid-crystalline textures.

Abstract: The single cluster covering approach provides a plausible mechanism for the formation and stability of octagonal and decagonal quasiperiodic structures. For dodecagonal quasiperiodic patterns, such a single cluster covering scheme is still unavailable. We demonstrate that ship tiling, one of the dodecagonal quasiperiodic structures, can be completely covered by a single cluster. A deflation procedure is devised by assigning proper orientations to different tiles, and nine types of vertex configurations, if the mirror patterns are considered to be identical, have been identified, which fulfill the closure condition under deflation and all result in a T-cluster centered at the vertex.

Abstract: We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Benard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle $${\pi/q, q \geqq 4}$$ . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.

Abstract: We study the dynamics of macroscopically coherent matter waves of an ultracold atomic spin-1 or spinor condensate on a ring lattice of six sites and demonstrate the spatiotemporal internal Josephson effect. Using a discrete solitary mode of uncoupled spin components as an initial condition, the time evolution of this many-body system is found to be characterized by two dominant frequencies leading to quasiperiodic dynamics at various sites. The dynamics of spatially averaged and spin-averaged degrees of freedom, however, is periodic enabling a unique identification of the two frequencies. By increasing the spin-dependent atom-atom interaction strength we observe a resonance state, where the ratio of the two frequencies is a characteristic integer multiple and the spin-and-spatial degrees of freedom oscillate in ``unison''. Crucially, this resonant state is found to signal the onset of chaotic dynamics characterized by a broadband spectrum. In a ferromagnetic spinor condensate with attractive spin-dependent interactions, the resonance is accompanied by a transition from oscillatory- to rotational-type dynamics as the time evolution of the relative phase of the matter wave of the individual spin projections changes from bounded to unbounded.

Abstract: We performed measurements at helium temperatures of the electronic transport in an InAs quantum wire ($R_{wire} \sim 30$\,k$\Omega$) in the presence of a charged tip of an atomic force microscope serving as a mobile gate. The period and the amplitude of the observed quasiperiodic oscillations are investigated in detail as a function of electron concentration in the linear and non-linear regime. We demonstrate the influence of the tip-to-sample distance on the ability to locally affect the top subband electrons as well as the electrons in the disordered sea. Furthermore, we introduce a new method of detection of the subband occupation in an InAs wire, which allows us to evaluate the number of the electrons in the conductive band of the wire.

Abstract: We examine the connectivity fluctuations across networks obtained when the horizontal visibility (HV) algorithm is used on trajectories generated by nonlinear circle maps at the quasiperiodic transition to chaos. The resultant HV graph is highly anomalous as the degrees fluctuate at all scales with amplitude that increases with the size of the network. We determine families of Pesin-like identities between entropy growth rates and generalized graph-theoretical Lyapunov exponents. An irrational winding number with pure periodic continued fraction characterizes each family. We illustrate our results for the so-called golden, silver, and bronze numbers.

Abstract: The design of periodic and quasiperiodic structures possessing innovative filtering properties for elastic waves opens the way to the realization of elastic metamaterials. In these structures prestress has a strong influence, ‘shifting’ in frequency, but also ‘annihilating’ or ‘nucleating’ band gaps. The effects of prestress are demonstrated with examples involving flexural waves in periodic and quasiperiodic beams and periodic plates. Results highlight that prestress can be employed as a ‘tuning parameter’ for continuously changing vibrational properties of elastic metamaterials.