Abstract: One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-Andre (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

Abstract: We present both a theoretical description and experimental observation of the nonlinear mutual interactions between a pair of copropagative breathers in the framework of the focusing one-dimensional nonlinear Schrodinger equation. As a general case, we show that the resulting bound state of breathers exhibits moleculelike behavior with quasiperiodic oscillatory dynamics (i.e., internal coherent interactions and pulsations), while for commensurate conditions the molecule oscillations become exactly periodic. Our theoretical model is confirmed by an experimental observation of shaped moleculelike breather light waves propagating in a nearly conservative optical fiber system. Our work sheds new light on the existence of localized wave structures and recurrence dynamics beyond the multisoliton complexes.

Abstract: Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the so-called self-dual symmetry usually display a mobility edge, similarly as truly disordered systems in a dimension strictly higher than two. Here, we determine the critical localization properties of single particles in shallow, one-dimensional, quasiperiodic models and relate them to the fractal character of the energy spectrum. On the one hand, we determine the mobility edge and show that it separates the localized and extended phases, with no intermediate phase. On the other hand, we determine the critical potential amplitude and find the universal critical exponent ν≃1/3. We also study the spectral Hausdorff dimension and show that it is nonuniversal but always smaller than unity, hence showing that the spectrum is nowhere dense. Finally, applications to ongoing studies of Anderson localization, Bose-glass physics, and many-body localization in ultracold atoms are discussed.

Abstract: Topological boundary and interface modes are generated in an acoustic waveguide by simple quasiperiodic patterning of the walls The procedure opens many topological gaps in the resonant spectrum and qualitative as well as quantitative assessments of their topological character are supplied In particular, computations of the bulk invariant for the continuum wave equation are performed The experimental measurements reproduce the theoretical predictions with high fidelity In particular, acoustic modes with high $Q$ factors localized in the middle of a breathable waveguide are engineered by a simple patterning of the walls

Abstract: We study the localization transitions for coupled one-dimensional lattices with quasiperiodic potential. In addition to the localized and extended phases, there is an intermediate mixed phase that can be easily explained decoupling the system so as to deal with effective uncoupled Aubry-Andr\'e chains with different transition points. We clarify, therefore, the origin of such an intermediate phase, finding the conditions for getting a uniquely defined mobility edge for such coupled systems. Finally, we consider many coupled chains with an energy shift that compose an extension of the Aubry-Andr\'e model in two dimensions. We study the localization behavior in this case comparing the results with those obtained for a truly aperiodic two-dimensional (2D) Aubry-Andr\'e model, with quasiperiodic potentials in any directions, and the 2D Anderson model.

Abstract: Few-level quantum systems driven by ${n}_{\mathrm{f}}$ incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in ${n}_{\mathrm{f}}$ spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. We organize the fundamental solutions of the frequency lattice model for ${n}_{\mathrm{f}}=2$ into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.

Abstract: This paper identifies a new dynamical phase at intermediate quasiperiodic potential. This phase, denoted as S phase, is characterized by power-law like information spreading and large fluctuations in the eigenstate entanglement, distinct from the thermal, localized, phase at weak and strong potentials and shown to be potentially responsible for the slow dynamics observed in cold-atom experiments.

Abstract: The many-body localization transition in quasiperiodic systems has been extensively studied in recent ultracold atom experiments. At intermediate quasiperiodic potential strength, a surprising Griffiths-like regime with slow dynamics appears in the absence of random disorder and mobility edges. In this work, we study the interacting Aubry-Andre model, a prototype quasiperiodic system, as a function of incommensurate potential strength using a novel dynamical measure, information scrambling, in a large system of 200 lattice sites. Between the thermal phase and the many-body localized phase, we find an intermediate dynamical phase where the butterfly velocity is zero and information spreads in space as a power-law in time. This is in contrast to the ballistic spreading in the thermal phase and logarithmic spreading in the localized phase. We further investigate the entanglement structure of the many-body eigenstates in the intermediate phase and find strong fluctuations in eigenstate entanglement entropy within a given energy window, which is inconsistent with the eigenstate thermalization hypothesis. Machine-learning on the entanglement spectrum also reaches the same conclusion. Our large-scale simulations suggest that the intermediate phase with vanishing butterfly velocity could be responsible for the slow dynamics seen in recent experiments.

Abstract: We report some organized structures of two linearly coupled logistic maps with different harvesting. The coupled system exhibits chaos via period-bubbling and quasiperiodic routes for identical and weak coupling strength, in contrast to conventional period-doubling route for a simple logistic map. Studies reveal the existence of infinite families of periodic Arnold tongues and self-similar shrimp-shaped structures with period-adding sequences for periodic windows embedded in quasiperiodic and chaotic regions, respectively. Different Fibonacci-like sequences are formed leading to the Golden Mean. The shrimp-shaped structures maintain period 3-times self-similarity scaling. The quasiperiodicity route is the necessary condition for the occurrence of periodic Arnold tongues in this coupled system resulting in the appearance of shrimps in the chaotic region near the tongues. It is also revealed that the existence of shrimp implies the period-bubbling cascade but the reverse is not true. The bifurcation-induced hysteresis is born in a certain parameter range resulting in the birth of coexisting multiple attractors of different kinds. Basin sets of the coexisting attractors have either self-similar or intertwining fractal basin boundaries.

Abstract: We discuss a two-dimensional system under the perturbation of a moir\'e potential, which takes the same geometry and lattice constant as the underlying lattices but mismatches up to relative rotation. Such a self-dual model belongs to the orthogonal class of a quasiperiodic system whose features have been evasive in previous studies. We find that such systems enjoy the same scaling exponent as the one-dimensional Aubry-Andr\'e model $\ensuremath{ u}\ensuremath{\approx}1$, which saturates the Harris bound $\ensuremath{ u}g2/d=1$ in two dimensions. Meanwhile, there exists a continuous and rapid change for the inverse participation ratio in the eigenstate-disorder plane, different from the typical one-dimensional situation where only a few or no steplike contours show up. An experimental scheme based on optical lattices is discussed. It allows for using lasers of arbitrary wavelengths and therefore is more applicable than the one-dimensional situations requiring laser wavelengths close to certain incommensurate ratios.

Abstract: We study high-temperature magnetization transport in a many-body spin-1/2 chain with on-site quasiperiodic potential governed by the Fibonacci rule. In the absence of interactions it is known that the system is critical with the transport described by a continuously varying dynamical exponent (from ballistic to localized) as a function of the on-site potential strength. Upon introducing weak interactions, we find that an anomalous noninteracting dynamical exponent becomes diffusive for any potential strength. This is borne out by a boundary-driven Lindblad dynamics as well as unitary dynamics, with agreeing diffusion constants. This must be contrasted to a random potential where transport is subdiffusive at such small interactions. Mean-field treatment of the dynamics for small $U$ always slows down the noninteracting dynamics to subdiffusion, and is therefore unable to describe diffusion in an interacting quasiperiodic system. Finally, briefly exploring larger interactions we find a regime of interaction-induced subdiffusive dynamics, despite the on-site potential itself having no ``rare regions.''

Abstract: There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Motivated by this, we study transport in the quasiperiodic, one-dimensional Aubry-Andre model and its generalizations to two and three dimensions. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with localized states, to a metal, with extended states having (a) ballistic transport (one dimension), (b) superdiffusive transport (two dimensions), or (c) diffusive transport (three dimensions); precisely at the transition, the system displays subdiffusive critical states. We calculate the $\ensuremath{\beta}$ function $\ensuremath{\beta}(g)=dln(g)/dln(L)$ and show that, in one and two dimensions, single-parameter scaling is unable to describe the transition. Furthermore, the conductances show strong nonmonotonic variations with $L$ and an intricate structure of resonant peaks and subpeaks. In one dimension the positions of these peaks can be related precisely to the properties of the number that characterizes the quasiperiodicity of the potential; and the $L$ dependence of the Thouless conductance is multifractal. We find that, as dimension increases, this nonmonotonic dependence of $g$ on $L$ decreases and, in three dimensions, our results for $\ensuremath{\beta}(g)$ are reasonably well approximated by single-parameter scaling.

Abstract: Nowadays, a novelty of devices that use magnetic restoring forces to generate oscillations has increased substantially. These kinds of devices have been commonly used to energy harvesting area. Therefore, in this paper, numerical and analytical analyses of a non-ideal magnetic levitation system are carried out. The mathematical modeling of the magnetic levitation device is developed and examined considering an electrodynamical shaker to base-excite the main system, which is a non-ideal excitation. The magnetic levitation system has the form of a Duffing oscillator; thus, the nonlinear analysis is required to investigate the energy harvesting potential of this nonlinear system. The novelty here is the use of the shaker to the excitation which is non-ideal. The method of multiple scales is applied to investigate the modes of vibration of the coupled system, which will remark the non-ideality and nonlinear phenomena of the system. The average harvested power is described by through expressions related to the coupling between the mechanical and electrical domains. Moreover, it was developed an expression for the excitation frequency where the maximum harvested power is obtained. The results were obtained based on the numerical method of Runge–Kutta of fourth order with fixed step whose results are shown through phase planes, Poincare maps and parametrical variation. Such results showed multiple existence of behaviors (periodic, quasiperiodic and chaos), coexistence of attractors in a high sensibility of the initial conditions and interesting results of the maximum average power, obtaining high and continuous amount of energy in periodic and chaotic regions.

Abstract: We study an integrable system that is reducible to free fermions by a Jordan-Wigner transformation which is subjected to a Fibonacci driving protocol based on two noncommuting Hamiltonians. In the high-frequency limit $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, we show that the system reaches a nonequilibrium steady state, up to some small fluctuations which can be quantified. For each momentum $k$, the trajectory of the stroboscopically observed state lies between two concentric circles on the Bloch sphere; the circles represent the boundaries of the small fluctuations. The residual energy is found to oscillate in a quasiperiodic way between two values which correspond to the two Hamiltonians that define the Fibonacci protocol. These results can be understood in terms of an effective Hamiltonian which simulates the dynamics of the system in the high-frequency limit.

Abstract: We study periodic, quasiperiodic (Thue-Morse, Fibonacci, period doubling, Rudin-Shapiro), fractal (Cantor, generalized Cantor), Kolakoski, and random binary sequences using a tight-binding wire model, where a site is a monomer (e.g., in DNA, a base pair). We use B-DNA as our prototype system. All sequences have purines, guanine (G) or adenine (A), on the same strand, i.e., our prototype binary alphabet is {G,A}. Our aim is to examine the influence of sequence intricacy and magnitude of parameters on energy structure, localization, and charge transport. We study quantities such as autocorrelation function, eigenspectra, density of states, Lyapunov exponents, transmission coefficients, and current-voltage curves. We show that the degree of sequence intricacy and the presence of correlations decisively affect the aforementioned physical properties. Periodic segments have enhanced transport properties. Specifically, in homogeneous sequences transport efficiency is maximum. There are several deterministic aperiodic sequences that can support significant currents, depending on the Fermi level of the leads. Random sequences is the less efficient category.

Abstract: In this paper, we have employed the transfer-matrix method to study theoretically the light waves propagation in extrinsic magnetized plasma multilayer, which is composed of a bulk plasma system influenced by the presence of spatially varying external magnetic field, which leads to a photonic bandgap device. The multilayered structures are arranged in periodic, quasiperiodic (Fibonacci, Octonacci, Thue–Morse, and double period), and Gaussian random fashions. The numerical results show the emergence of two main photonic bandgaps: the first gap for low frequencies and the second one for higher frequencies. We investigate the robust nature of the higher frequencies bandgap since it shows up to be invariant to different values of applied external magnetic fields and electron density as well as changes in the position and thickness of the layers introduced by the quasiperiodic and the Gaussian random sequences, respectively. The most surprising result is that this desired robust bandgap is broadening without any intermediate resonant peaks while the randomness in the layer thickness is introduced, which had not been observed in previous works about this same system.

Abstract: We theoretically study the dynamics of the ground-state cooling and the quantum entanglement in a modulated optomechanical system, where the frequency of the mechanical oscillator and the optical field or the strength of the driving laser is time dependent. In this paper, we focus mainly on the fact that the system works in the regime of blue detuning. It is found that in the long-time limit the steady-state phonon number can be decreased significantly so that the mechanical oscillator is cooled to the ground state by appropriately selecting the frequency and the amplitude of the modulation. Further, with the help of the dynamical modulation in the system, the time evolution of the entanglement preevaluation value displays the death and rebirth of quantum entanglement between the mechanical oscillator and the optical field, which correspond, respectively, to the increase and decrease of the effective phonon number of the mechanical oscillator. In particular, when the steady-state dynamics of the system is quasiperiodic, the dynamics of the entanglement preevaluation value exhibits a quasiperiodic behavior, which means that the quantum phenomena can be represented in nonlinear classical dynamics.

Abstract: We investigate a generalized Aubry-Andr\'e-Harper (AAH) model with $p$-wave superconducting pairing. Both the hopping amplitudes between the nearest-neighboring lattice sites and the on-site potentials in this system are modulated by a cosine function with a periodicity of $1/\ensuremath{\alpha}$. In the incommensurate case $[\ensuremath{\alpha}=\left(\sqrt{5}\ensuremath{-}1\right)/2]$, due to the modulations on the hopping amplitudes, the critical region of this quasiperiodic system is significantly reduced and the system becomes easier to be turned from extended states to localized states. In the commensurate case $(\ensuremath{\alpha}=1/2)$, we find that this model shows three different phases when we tune the system parameters: Su-Schrieffer-Heeger (SSH)-like trivial, SSH-like topological, and Kitaev-like topological phases. The phase diagrams and the topological quantum numbers for these phases are presented in this work. This generalized AAH model combined with superconducting pairing provides us with a useful test field for studying the phase transitions from extended states to Anderson localized states and the transitions between different topological phases.

Abstract: We investigated the lifetime of spin wave eigenmodes in periodic and quasiperiodic sequences of Py and Co wires. Those materials differ significantly in damping coefficients, therefore, the spatial distribution of the mode amplitude within the structure is important for the lifetime of collective spin wave excitations. Modes of the lower frequencies prefer to concentrate in Py wires, because of the lower FMR frequency for this material. This inhomogeneous distribution of amplitude of modes (with lower amplitude in material of higher damping and with higher amplitude in material of lower damping) is preferable for extending the lifetime of the collective excitations beyond the volume average of lifetimes for solid materials. We established the relation between the profile of the mode and its lifetime for periodic and quasiperiodic structures. We performed also the comparative studies in order to find the differences resulting from complexity of the structure and enhancement of localization in quasiperiodic system on the lifetime of spin waves.

Abstract: Understanding singularities in ordered structures, such as dislocations in lattice modulation and solitons in charge ordering, offers great opportunities to disentangle the interactions between the electronic degrees of freedom and the lattice. Specifically, a modulated structure has traditionally been expressed in the form of a discrete Fourier series with a constant phase and amplitude for each component. Here, we report atomic scale observation and analysis of a new modulation wave in hole-doped LuFe_{2}O_{4+δ} that requires significant modifications to the conventional modeling of ordered structures. This new modulation with an unusual quasiperiodic singularity can be accurately described only by introducing a well-defined secondary modulation vector in both the phase and amplitude parameter spaces. Correlated with density-functional-theory (DFT) calculations, our results reveal that those singularities originate from the discontinuity of lattice displacement induced by interstitial oxygen in the system. The approach of our work is applicable to a wide range of ordered systems, advancing our understanding of the nature of singularity and modulation.

Abstract: In this paper, we report on the dynamics of a discrete-time mathematical model, which is obtained by the forward Euler method from the continuous-time mathematical model of the glycolysis process. More specifically, here we investigate the parameter-space of a two-dimensional map resulting from this discretization process. Different places where period-doubling and Naimark–Sacker bifurcations occur are determined. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region which is a result of a Naimark–Sacker bifurcation. We identify period-adding, Farey, and Fibonacci sequences of periodic structures embedded in this quasiperiodic region.

Abstract: Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Benard–Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle $${\pi/q}$$ . There is a small divisor problem for $${q \geqq 4}$$ . Using the results of Berti–Bolle–Procesi in 2010, we adapt it to a Navier–Stokes system ruling the Benard–Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift–Hohenberg PDE model. For using the Nash–Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point. There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of L(N)L(N)* which depends mainly quadratically on the main parameter. A careful study of the “bad set”of parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti et al. for solving ”the range equation”. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional “bifurcation equation. For any $${q \geqq 4}$$ , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number.

Abstract: This article is a survey of the Novikov problem of the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold M. Equivalently, this is to the study of the level sets of multivalued functions on M. To date, this problem was thoroughly investigated only for \(M={\mathbb {T}}^n\) and multivalued maps \(F:{\mathbb {T}}^n\to \mathbb {R}^{n-1}\) in three different particular cases: when all components of F but one are multivalued, started by Novikov in 1982; when all components of F but one are singlevalued, started by Zorich in 1994; when none of the components is singlevalued, started by Arnold in 1991. The first two problems can be formulated as the study of the level sets of certain quasiperiodic functions, the last as level sets of pseudoperiodic functions. In this survey we present the main analytical and numerical results to date and some physical phenomena where they play a fundamental role.

Abstract: The nearest-neighbor Aubry–Andre quasiperiodic localization model is generalized to include power-law translation-invariant hoppings Tl ∝ t/la or power-law Fourier coefficients Wm ∝ w/mb in the qua...

Abstract: At present chaotic vibrations are the one of the most interesting and explored subjects in nonlinear dynamics. Particularly the routes to chaos in non-smooth dynamical systems are of the special scientists’ interest. Quasiperiodic route to chaos in nonlinear non-smooth discontinuous 2-DOF vibroimpact system is studied in this paper. In narrowfrequency range different oscillatory regimes have succeeded each other many times under very small control parameter varying. There were periodic subharmonic regimes - chatters, quasiperiodic, and chaotic regimes. There were the zones of transition from one regime to another, the zones of prechaotic and postchaotic motion. The hysteresis effects (jump phenomena) occurred for increasing and decreasing frequencies. The chaoticity of obtained regime has been confirmed by typical views of Poincare map and Fourier spectrum, by the positive value of the largest Lyapunov exponent, and by the fractal structure of Poincare map.