Abstract: We study the Schrodinger equation on $\R$ with a potential behaving as $x^{2l}$ at infinity, $l\in[1,+\infty)$ and with a small time quasiperiodic perturbation. We prove that, if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.

Abstract: We demonstrate, through three-dimensional discrete dislocation dynamics simulations, that the complex dynamical response of nano- and microcrystals to external constraints can be tuned. Under load rate control, strain bursts are shown to exhibit scale-free avalanche statistics, similar to critical phenomena in many physical systems. For the other extreme of displacement rate control, strain burst response transitions to quasiperiodic oscillations, similar to stick-slip earthquakes. External load mode control is shown to enable a qualitative transition in the complex collective dynamics of dislocations from self-organized criticality to quasiperiodic oscillations.

Abstract: We investigated the lifetime of spin wave (SW) eigenmodes in periodic and quasiperiodic sequences of Py and Co wires. These materials differ significantly in damping coefficients, therefore, the spatial distribution of the mode's amplitude within the structure is important for the lifetime of collective SW excitations. Modes of the lower frequencies prefer to concentrate in Py wires, because of the lower ferromagnetic resonance (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 also performed comparative studies in order to find the differences resulting from complexity of the structure and enhancement of localization in the quasiperiodic system on the lifetime of SWs.

Abstract: We use the Floquet-Bloch transform to reduce variational formulations of surface scattering problems for the Helmholtz equation from periodic and locally perturbed periodic surfaces to equivalent variational problems formulated on bounded domains. To this end, we establish various mapping properties of that transform between suitable weighted Sobolev spaces on periodic strip-like domains and coupled families of quasiperiodic Sobolev spaces. Our analysis shows in particular that the decay of solutions to surface scattering problems from locally perturbed periodic surfaces is precisely characterized by the smoothness of its Bloch transform in the quasiperiodicity.

Abstract: We establish localization type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with quasiperiodic potentials defined by piecewise Holder functions.

Abstract: Estimates of the black hole mass $M$ and dimensionless spin $a$ in the microquasar GRO J1655-40 implied by strong gravity effects related to the timing and spectral measurements are controversial, if the mass restriction determined by the dynamics related to independent optical measurements, $M_{\rm opt}=(5.4\pm0.3) M_{\odot}$, are applied. The timing measurements of twin high-frequency (HF) quasiperiodic oscillations (QPOs) with frequency ratio $3:2$ and the simultaneously observed low-frequency (LF) QPO imply the spin in the range $a\in(0.27-0.29)$ if models based on the frequencies of the geodesic epicyclic motion are used to fit the timing measurements, and correlated creation of the twin HF QPOs and the LF QPO at a common radius is assumed. On the other hand, the spectral continuum method implies $a\in(0.65-0.75)$, and the Fe-line-profile method implies $a\in(0.94-0.98)$. This controversy can be cured, if we abandon the assumption of the occurrence of the twin HF QPOs and the simultaneously observed LF QPO at a common radius. We demonstrate that the epicyclic resonance model of the twin HF QPOs is able to predict the spin in agreement with the Fe-profile method, but no model based on the geodesic epicyclic frequencies can be in agreement with the spectral continuum method. We also show that the non-geodesic string loop oscillation model of twin HF QPOs predicts spin $a>0.3$ under the optical measurement limit on the black hole mass, in agreement with both the spectral continuum and Fe-profile methods.

Abstract: Bifurcations and chaotic behaviors of dust acoustic traveling waves in magnetoplasmas with nonthermal ions featuring Cairns–Tsallis distribution is investigated on the framework of the further modified Kadomtsev–Petviashili (FMKP) equation. The FMKP equation is derived employing the reductive perturbation technique (RPT). Bifurcations of dust acoustic traveling waves of the FMKP equation is presented. Using the bifurcation theory of planar dynamical systems, two new analytical traveling wave solutions for solitary and periodic waves are derived depending on the parameters $$\alpha , \alpha _1, q, l$$ and U. Considering an external periodic perturbation, the chaotic behavior of dust acoustic traveling waves is investigated through quasiperiodic route to chaos. The parameter q significantly affects the chaotic behavior of the perturbed FMKP equation.

Abstract: In this work, the concept of quantum Fisher information (QFI) is used to characterize the quantum transitions and factorization transitions in one-dimensional anisotropic XY models with periodic coupling interaction and quasiperiodic one. For the periodic-two model, it is found that the Ising transition and anisotropic transition can be distinctively illustrated by the evolution of QFI and its first-order derivatives, confirmed additionally by the scaling behavior. For the quasiperiodic Fibonacci chain, the number of quantum phase transitions increases from one to the lth Fibonacci number $$F_{l}$$Fl when the anisotropic parameter $$\gamma $$? approaches zero. The phase diagram for the approximant $$F_{l}=8 $$Fl=8 is derived as an example. In addition, the factorization transition in the two models can be marked by the correlation quantity defined from the QFI. The present work demonstrates the implication of the QFI as a general fingerprint to characterize the quantum transitions and factorization transitions.

Abstract: We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincare-type inequality.

Abstract: We describe a way to obtain a two-dimensional quasiperiodic tiling with eight-fold symmetry using cold atoms. A series of such optical tilings, related by scale transformations, is obtained for a series of specific values of the chemical potential of the atoms. A theoretical model for the optical system is described and compared with that of the well-known cut-and-project method for the Ammann-Beenker tiling. This type of cold atom structure should allow the simulation of several important lattice models for interacting quantum particles and spins in quasicrystals.

Abstract: The theory of Kolmogorov, Arnold, and Moser (KAM) consists of a set of results regarding the persistence of quasiperiodic solutions, primarily in Hamiltonian systems.We bring forward a “twisted conjugacy” normal form, due to Herman, which contains all the (not so) hard analysis. We focus on the real analytic setting. A variety of KAM results follow, includingmost classical statements as well asmore general ones. This strategy makes it simple to deal with various kinds of degeneracies and symmetries. As an example of application, we prove the existence of quasiperiodic motions in the spatial lunar three-body problem.

Abstract: Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we firstly rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we secondly transform this problem into a coupled family of quasiperiodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasiperiodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasiperiodicity's unit cells. We also provide a simple and efficient way for implementation that does not require numerical integration in the quasiperiodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme.

Abstract: Heterotic supergravity with (1+3)--dimensional domain wall configurations and (warped) internal, six dimensional, almost-Kahler manifolds $\ ^6\mathbf{X}$ are studied. Considering ten dimensional spacetimes with nonholonomic distributions and conventional double fibrations, 2+2+...=2+2+3+3, and associated $SU(3)$ structures on internal space, we generalize for real, internal, almost symplectic gravitational structures the constructions with gravitational and gauge instantons of tanh-kink type. They include the first $\alpha ^{\prime}$ corrections to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the Yang-Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of solutions depending on the type of nonholonomic distributions and deformations of 'prime' instanton configurations characterized by two real supercharges. This corresponds to $\mathcal{N}=1/2$ supersymmetric, nonholonomic manifolds from the four dimensional point of view. Our method provides a unified description of embedding nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first order $\alpha ^{\prime}$ corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped almost-Kahler configurations, which allows us to generate nontrivial (1+3)-d domain walls and black hole deformations determined by quasiperiodic internal space structures. This formalism is utilized in our associated publication [EPJC 77 (2017) 17, arXiv: 1608.01980] in order to construct and study generic off-diagonal nonholonomic deformations of the Kerr metric, encoding contributions from heterotic supergravity.

Abstract: A very serious concern of scientists dealing with crystal structure refinement, including theoretical research, pertains to the characteristic bias in calculated versus measured diffraction intensities, observed particularly in the weak reflection regime. This bias is here attributed to corrective factors for phonons and, even more distinctly, phasons, and credible proof supporting this assumption is given. The lack of a consistent theory of phasons in quasicrystals significantly contributes to this characteristic bias. It is shown that the most commonly used exponential Debye–Waller factor for phasons fails in the case of quasicrystals, and a novel method of calculating the correction factor within a statistical approach is proposed. The results obtained for model quasiperiodic systems show that phasonic perturbations can be successfully described and refinement fits of high quality are achievable. The standard Debye–Waller factor for phonons works equally well for periodic and quasiperiodic crystals, and it is only in the last steps of a refinement that different correction functions need to be applied to improve the fit quality.

Abstract: We consider the electric conductivity in normal metals in presence of a strong magnetic field. It is assumed here that the Fermi surface of a metal has rather complicated form such that different types of quasiclassical electron trajectories can appear on the Fermi level for different directions of B. The effects we consider are connected with the existence of regular (stable) open electron trajectories which arise in general on complicated Fermi surfaces. The trajectories of this type have a nice geometric description and represent quasiperiodic lines with a fixed mean direction in the p-space. Being stable geometric objects, the trajectories of this kind exist for some open regions in the space of directions of B, which can be represented by "Stability Zones" on the unit sphere. The main goal of the paper is to give a description of the analytical behavior of conductivity in the Stability Zones, which demonstrates in general rather nontrivial properties.

Abstract: The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects gives rise to rich dynamics: this extends from Poincare-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections or to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, and chaotic spatiotemporal structures. Suitable low-dimensional phase-space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

Abstract: We investigate the transmission properties of quasiperiodic or aperiodic structures based on graphene arranged according to the Cantor sequence. In particular, we have found self-similar behaviour in the transmission spectra, and most importantly, we have calculated the scalability of the spectra. To do this, we implement and propose scaling rules for each one of the fundamental parameters: generation number, height of the barriers and length of the system. With this in mind we have been able to reproduce the reference transmission spectrum, applying the appropriate scaling rule, by means of the scaled transmission spectrum. These scaling rules are valid for both normal and oblique incidence, and as far as we can see the basic ingredients to obtain self-similar characteristics are: relativistic Dirac electrons, a self-similar structure and the non-conservation of the pseudo-spin.

Abstract: This thesis is concerned with dynamics of conservative nonlinear waves on bounded domains. In general, there are two scenarios of evolution. Either the solution behaves in an oscillatory, quasiperiodic manner or the nonlinear effects cause the energy to concentrate on smaller scales leading to a turbulent behaviour. Which of these two possibilities occurs depends on a model and the initial conditions. In the quasiperiodic scenario there exist very special time-periodic solutions. They result for a delicate balance between dispersion and nonlinear interaction. The main body of this dissertation is concerned with construction (by means of perturbative and numerical methods) of time-periodic solutions for various nonlinear wave equations on bounded domains. While turbulence is mainly associated with hydrodynamics, recent research in General Relativity has also revealed turbulent phenomena. Numerical studies of a self-gravitating massless scalar field in spherical symmetry gave evidence that anti-de Sitter space is unstable against black hole formation. On the other hand there appeared many examples of asymptotically anti-de Sitter solutions which evade turbulent behaviour and appear almost periodic for long times. We discuss here these two contrasting scenarios putting special attention to the construction and properties of strictly time-periodic solutions. We analyze different models where solutions of this type exist. Moreover, we describe similarities and differences among these models concerning properties of time-periodic solutions and methods used for their construction.

Abstract: This paper reports on an investigation of the two-dimensional parameter-space of a generalized Nose–Hoover oscillator. It is a mathematical model of a thermostated harmonic oscillator, which consists of a set of three autonomous first-order nonlinear ordinary differential equations. By using Lyapunov exponents to numerically characterize the dynamics of the model at each point of this parameter-space, it is shown that dissipative quasiperiodic structures are present, embedded in a chaotic region. The same parameter-space is also used to confirm the multistability phenomenon in the investigated mathematical model.

Abstract: We study the magnetic response and valence fluctuations in the extended Anderson model on a two-dimensional Penrose lattice using real-space dynamical mean-field theory combined with the continuous-time quantum Monte Carlo method. Calculating the f-electron number, c–f spin correlations, and magnetic susceptibility at each site, we find site-dependent formation of the singlet state and valence distributions at low temperatures, which are reflected by the quasiperiodic lattice structure. The bulk magnetic susceptibility is also addressed.

Abstract: A tight-binding model of a multi-leg ladder network with a continuous quasiperiodic modulation in both the site potential and the inter-arm hopping integral is considered. The model mimics optical lattices where ultra-cold fermionic or bosonic atoms are trapped in double well potentials. It is observed that, the relative phase difference between the on-site potential and the inter-arm hopping integral, which can be controlled by the tuning of the interfering laser beams trapping the cold atoms, can result in a mixed spectrum of one or more absolutely continuous subband(s) and point like spectral measures. This opens up the possibility of a re-entrant metal-insulator transition. The subtle role played by the relative phase difference mentioned above is revealed, and we corroborate it numerically by working out the multi-channel electronic transmission for finite two-, and three-leg ladder networks. The extension of the calculation beyond the two-leg case is trivial, and is discussed in the work.

Abstract: A map on a torus is called “quasiperiodic” if there is a change of variables which converts it into a pure rotation in each coordinate of the torus. We develop a numerical method for finding this change of variables, a method that can be used effectively to determine how smooth (i.e., differentiable) the change of variables is, even in cases with large nonlinearities. Our method relies on fast and accurate estimates of limits of ergodic averages. Instead of uniform averages that assign equal weights to points along the trajectory of N points, we consider averages with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint N∕2. We provide a one-dimensional quasiperiodic map as an example and show that our weighted averages converge far faster than the usual rate of O(1∕N), provided f is sufficiently differentiable. We use this method to efficiently numerically compute rotation numbers, invariant densities, conjugacies of quasiperiodic systems, and to provide evidence that the changes of variables are (real) analytic.

Abstract: It is well acknowledged that the sequence of glacial-interglacial cycles is paced by the astronomical forcing. However, how much is the sequence robust against natural fluctuations associated, for example, with the chaotic motions of atmosphere and oceans? In this article, the stability of the glacial-interglacial cycles is investigated on the basis of simple conceptual models. Specifically, we study the influence of additive white Gaussian noise on the sequence of the glacial cycles generated by stochastic versions of several low-order dynamical system models proposed in the literature. In the original deterministic case, the models exhibit different types of attractors: a quasiperiodic attractor, a piecewise continuous attractor, strange nonchaotic attractors, and a chaotic attractor. We show that the combination of the quasiperiodic astronomical forcing and additive fluctuations induces a form of temporarily quantised instability. More precisely, climate trajectories corresponding to different noise realizations generally cluster around a small number of stable or transiently stable trajectories present in the deterministic system. Furthermore, these stochastic trajectories may show sensitive dependence on very small amounts of perturbations at key times. Consistently with the complexity of each attractor, the number of trajectories leaking from the clusters may range from almost zero (the model with a quasiperiodic attractor) to a significant fraction of the total (the model with a chaotic attractor), the models with strange nonchaotic attractors being intermediate. Finally, we discuss the implications of this investigation for research programmes based on numerical simulators.

Abstract: We study valence fluctuations in the extended Anderson model on two-dimensional Penrose lattice, using the real-space dynamical mean-field theory combined with the continuous-time Monte Carlo method. Calculating $f$-electron number, $c$-$f$ spin correlations, and magnetic susceptibility at each site, we find site-dependent formations of the singlet state and valence distribution at low temperatures, which are reflected by the quasiperiodic lattice structure. The bulk magnetic susceptibility is also addressed.

Abstract: This paper involves the dynamics of a delay limit cycle oscillator being driven by a time-varying perturbation in the delay: x=−x(t−T(t))−ϵx3 with delay T(t)=π2+ϵk+ϵcosωt. This delay is chosen to periodically cross the stability boundary for the x=0 equilibrium in the constant-delay system. For most of parameter space, the system is non-resonant, leading to quasiperiodic behavior. However, a region of 2:1 resonance is shown to exist where the system׳s response frequency is entrained to half of the forcing frequency ω. By a combination of analytical and numerical methods, we find that the transition between quasiperiodic and entrained behavior consists of a variety of local and global bifurcations, with corresponding regions of multiple stable and unstable steady-states.

Abstract: In this PhD thesis we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty }$-reducible cocycles are dense in the $C^{\infty }$ topology, for a full measure set of frequencies. We will firstly define two invariants of the dynamics, which we will call energy and degree and which give a preliminary distinction between reducible and non-reducible cocycles. We will then take up the proof of the density theorem. We will show that an algorithm of renormalization converges to perturbations of simple models, indexed by the degree. Finally, we will analyse these perturbations using methods inspired by K.A.M. theory. In this context we will prove that if a $C^{\infty }$ cocycle is measurably reducible to a diophantine constant, it is actually $C^{\infty }$-reducible.