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The cell mapping method is one of the most powerful tools in the global analysis of nonlinear dynamical systems. It accurately depicts the boundaries of the domains of attractors but is not the ideal method for diagnosing the characteristics of attractors. To improve the performance of the conventional cell mapping method, an approach telling quasiperiodicity from chaos is developed in the current research. The proposed method uses the convergence speed of a probability distribution on the attractor as the criterion. For a chaotic system, the distribution converges quickly to the invariant density in several iterations. As for a quasiperiodic system, the distribution converges extremely slowly to the invariant density. The difference can be explained by the eigenvalues of the transition matrices of different systems. The transition matrix of a chaotic system has irregular eigenvalues, and the second-largest eigenvalue is small enough, while the matrix of a quasiperiodic system has regular eigenvalues, many of which are close to the unit circle. A theoretical analysis of some simple mappings is addressed to support this explanation. Furthermore, as a supplement to the analysis, a numerical method that calculates the Lyapunov exponent from the transition matrix is also discussed. [ABSTRACT FROM AUTHOR]

An innovative box-counting method is developed in this research for diagnozing the nonlinear characteristics of dynamical systems. With the method developed, an approach that depicts the evolutionary process on Poincaré maps is established such that the nonlinear dynamical characteristics of the transient and stable process of the system can be graphically and quantitatively identified. A Duffing–van der Pol system is adopted in the research to demonstrate an application of the method. A diagram graphically describing the periodic, quasiperiodic, chaotic, and transient chaotic regions of the system's responses is constructed based on the method. Furthermore, the nature of different box-point curves is explained based on the topology of chaos and quasiperiodicity. The method developed shows innovation and efficiency in diagnozing nonlinear dynamical systems based on the topological properties of general nonlinear systems. [ABSTRACT FROM AUTHOR]

Dimension‐reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost periodicity of the energies in the directions of the mid‐plane of the film. In this note, we consider thin films, obtained as sections of a periodic medium with a mid‐plane that may be incommensurate, that is, not containing periods other than 0. A geometric almost periodicity argument similar to the cut‐and‐project argument used for quasicrystals allows to prove a general homogenization result. [ABSTRACT FROM AUTHOR]

Background and Objectives: The Josephson effect is widely used for both generating and receiving very high frequency signals. The approaches and methods of nonlinear dynamics are commonly used: construction of phase portraits, bifurcation analysis, Kuramoto model approach, etc. Usually large arrays of identical junctions are considered. However, the non-identity of junctions leads to new and interesting effects. A very popular model is a chain of junctions connected via an RLC circuit. In this case, two types of non-identity are possible – in terms of critical currents IN through the junctions and in terms of the value rN of junction resistances. An increase in the number of junctions from two to three leads to the possibility of quasiperiodic dynamics with invariant tori of dimensions both two and three and to a complex structure of the parameter space. Materials and Methods: In this paper, we will mainly consider three junctions that are not identical in terms of resistance. As the main research tool, we will use the method of construction of Lyapunov exponent charts. Within the framework of this method, the type of dynamics of the system is determined by the signature of the spectrum of Lyapunov exponents. The parameter plane is scanned and the types of modes are identified at each point. The method is effective in that it allows one to study all types of possible regimes and fine details of the parameter space arrangement. Results: For the analysis of the dynamics of non-identical Josephson junctions, the method of Lyapunov exponent charts is effective. With its help, the regions of periodic regimes, regimes of two-frequency and three-frequency quasiperiodicity, chaos have been revealed. As a rule, the regions of two-frequency quasiperiodicity have the form of bands of different widths immersed in the region of three-frequency quasiperiodicity, which form the structure of the Arnold’s resonant web. Conclusion: The boundaries of the regions of two-frequency quasiperiodicity are the lines of saddle-node bifurcations of invariant tori. As the area of chaos increases, the Arnold’s resonant web can collapse. Changing the type of external circuit coupling the junctions does not fundamentally affect the dynamics of the system.

Analysis of the dynamics of chaotic systems through explicit analytical solutions has been a recent topic of research importance. The chaotic and synchronization dynamics of second-order non-autonomous electronic circuit systems has been studied analytically, while the analytical dynamics of third-order non-autonomous chaotic systems is yet to be studied. In this paper, we develop explicit analytical solutions for the state equations of a third-order, non-autonomous nonlinear electronic circuit system, namely the driven Chua's circuit. The dynamics of the system is studied using the analytical solutions for different set of system parameters, and the existence of the phenomena of multistability of attractors, period-doubling scenario, antimonotonicity and quasiperiodicity are reported. Analytical solutions are also developed to study the evolution of certain types of synchronization such as phase, phase-lag and complete synchronization observed in coupled non-identical and identical systems. Analytical studies on the chaotic and synchronization dynamics of a third-order, non-autonomous electronic circuit system are reported in the literature for the first time. [ABSTRACT FROM AUTHOR]

Background and Objectives: Quasi-periodic oscillations are widespread in nature and technology. In the phase space, quasi-periodic oscillations with a different number of incommensurable frequencies correspond to invariant tori of different dimensions. Multiparametric analysis of high-dimensional systems is quite difficult. The way out of this situation can be the transition from systems with continuous time to discrete maps. Materials and Methods: In the paper we use a discretization method for the transition from a continuous-time system (two coupled quasi-periodic generators) to a new high-dimensional map. The time derivatives are replaced by finite differences. There is a new additional parameter corresponding to the discretization step, with the variation of which the system can demonstrate new interesting properties. Results: A new high-dimensional map has been obtained by the discretization method of differential equation system of coupled quasi-periodic generators. For this map, charts of Lyapunov exponents have been constructed in the plane of the frequency detuning of generators and the coupling magnitude. The existence of invariant tori of different dimensions has been demonstrated. Graphs of Lyapunov exponents and Fourier spectra have been presented. The evolution of maps with an increase of the discretization parameter has been investigated, the destruction of high-dimensional tori has been demonstrated. The influence of noise of different intensity has been studied. Conclusion: The two-parameter Lyapunov analysis of the new high-dimensional map made it possible to identify regions of invariant tori of different dimensions, up to fivefrequency ones. The map demonstrates Fourier spectra characteristic of quasi-periodicity of increasing dimension. Quasi-periodic bifurcations of invariant tori and an Arnold resonance web based on invariant tori of different dimensions have been observed. With the growth of the discretization parameter, the destruction of high-dimensional tori occurs. An increase in the discretization parameter leads to the destruction of high-dimensional tori with an increase in the noise intensity.

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry-André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski-Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future. [ABSTRACT FROM AUTHOR]

We propose a spectral viewpoint for grain boundaries that are generally quasiperiodic. To accurately capture the spectra computationally, it is crucial to adopt the projection method for quasiperiodic functions. Armed with the Lifshitz-Petrich free energy, we take the spectral viewpoint to examine tilt grain boundaries of the hexagonal phase. Several ingredients of grain boundaries are extracted, which are not easy to obtain from real-space profiles. We find that only a few spectra substantially contribute to the formation of grain boundaries. Their linear relation to the intrinsic spectra of the bulk hexagonal phase is independent of the tilt angle. By examining the feature of the spectral intensities, we propose a definition of the interface width. The widths calculated from this definition are consistent with visual estimation. [ABSTRACT FROM AUTHOR]

Quasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text T is a string whose occurrences in T cover all positions of T. A seed of text T is a cover of a superstring of T. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in T with a small Hamming, Levenshtein or weighted edit distance. In previous work Sim et al. (J Korea Inf Sci Soc 29(1):16–21, 2002) and Christodoulakis et al. (J Autom Lang Comb 10(5/6), 609–626, 2005) showed that computing approximate covers and seeds, respectively, under weighted edit distance is NP-hard. They, therefore, considered restricted approximate covers and seeds which need to be factors of the original string T and presented polynomial-time algorithms for computing them. Further algorithms, considering approximate occurrences with Hamming distance bounded by k, were given in several contributions by Guth et al. They also studied relaxed approximate quasiperiods. We present more efficient algorithms for computing restricted approximate covers and seeds. In particular, we improve upon the complexities of many of the aforementioned algorithms, also for relaxed quasiperiods. Our solutions are especially efficient if the number (or total cost) of allowed errors is small. We also show conditional lower bounds for computing restricted approximate covers and prove NP-hardness of computing non-restricted approximate covers and seeds under the Hamming distance. [ABSTRACT FROM AUTHOR]

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a 'cut and project' procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients. [ABSTRACT FROM AUTHOR]

Influence of dust-neutral collisional frequency and nonextensivity on the linear and nonlinear dust-acoustic waves is studied in a magnetized collisional dusty plasma, which consists of negatively charged cold dust particles, q-nonextensive hot ions and hot electrons with stationary neutrals considering the dust-neutral collisions. It is observed through linear mode analysis that linear dust-acoustic wave behaves like a damping wave and its damping rate mainly depends on dust-neutral collisional frequency ( ν d n ). The damped Zakharov and Kuznetsov (ZK) equation is obtained employing the reductive perturbation technique. The effects of dust-neutral collisional frequency ( ν 0 ) and nonextensive parameter (q) are observed on the characteristic features of dust-acoustic solitary waves. Control of dynamic motions for dust-acoustic waves is investigated considering three-dimensional phase portraits, time-series analysis and sensitivity analysis. The parameters q i , q e and ν 0 play vital roles on the control of dynamic motions for dust-acoustic waves in presence of the external periodic force. Our study may be helpful to understand the behavior of dust-acoustic wave in the magnetosphere. [ABSTRACT FROM AUTHOR]

Quasiperiodic and chaotic features of the dust ion acoustic (DIA) waves in a magnetized dusty plasma is studied. Reductive perturbation technique (RPT) is used to derive the damped forced Kadomtsev–Petviashvili (KP) equation. The effect of the dust ion collisional frequency (νid0) on the quasiperiodic and chaotic motion of the DIA waves is studied. It is observed that in absence of dust–ion collisions, the DIA waves exhibit quasiperiodic behavior and the motion becomes chaotic while the dust–ion collisions are taken into consideration. Thus, the dust–ion collisional frequency (νid0) plays the role of switching parameter from quasiperiodic motion to chaotic motion for the DIA waves. [ABSTRACT FROM AUTHOR]

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. [ABSTRACT FROM AUTHOR]

Symmetry is not only used to simplify the analysis of three-phase electrical systems, but it is also used to define its voltages, currents and loads. When the loads are symmetrical, the currents of a three-phase AC system are expected to be symmetrical as well. Given the proper conditions, in converters such as the matrix converter (MC), the output voltages and currents are expected to be sinusoidal with periodic symmetry. However, in some cases this symmetry is broken so that, there appears nonlinear behaviors such as quasiperiodicity and so on. Based on simulations and experiments, this paper focuses on an analysis of a quasiperiodic behavior and the presence of a DC component in the output currents of a Venturini modulated MC. The presence of such behaviors in the output currents indicates that the symmetry in a period is broken. The broken symmetries appear when the input and output frequencies are mismatched. In addition, this paper shows the possibility to recover the symmetry of the output currents of the MC. The method for symmetry recovery is based on a time-delayed feedback control. The simulation and experimental results indicate the possibility of attenuating the quasiperiodic behavior and DC component. [ABSTRACT FROM AUTHOR]

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor. [ABSTRACT FROM AUTHOR]

Quasiperiodicity is a generalization of periodicity that was introduced in the early 1990s. Since then, dozens of algorithms for computing various types of quasiperiodicity were proposed. Our work is a step towards answering the general question: "Which algorithm for computing quasiperiods to choose?". The central notions of quasiperiodicity are covers and seeds. We implement algorithms for computing covers and seeds in the original and in new simplified versions and compare their efficiency on various types of data. We also discuss other known types of quasiperiodicity, distinguish partial covers as currently the most promising for large real-world data, and check their effectiveness using real-world data. [ABSTRACT FROM AUTHOR]

A model of quasiperiodic-chaotic neural networks is proposed on the basis of chaotic neural networks. A quasiperiodic-chaotic neuron exhibits quasiperiodic dynamics that an original chaotic neuron does not have. Quasiperiodic and chaotic solutions are exclusively isolated in the parameter space. The chaotic domain can be identified by the presence of a folding structure of an invariant closed curve. Using the property that the influence of perturbation is conserved in the quasiperiodic solution, we demonstrate short-term visual memory in which real numbers are acceptable for representing colors. The quasiperiodic solution is sensitive to dynamical noise when images are restored. However, the quasiperiodic synchronization among neurons can reduce the influence of noise. Short-term analog memory using quasiperiodicity is important in that it can directly store analog quantities. The quasiperiodic-chaotic neural networks are shown to work as large-scale analog storage arrays. This type of analog memory has potential applications to analog computation such as deep learning. [ABSTRACT FROM AUTHOR]

Using the definition of quasiperiodic function as a motivation, we introduce the idea of quasiperiodic music and detail the composition process of a quasiperiodic music piece, Raindrops in A minor. We also discuss connections between quasiperiodic music and other works of music theory and composition that make use of aperiodic order or periodic order with large periods, such as Lindenmayer systems, Vuza canons, Messiaen's Quatuor pour la Fin du Temps, and the phase music of Steve Reich. [ABSTRACT FROM AUTHOR]

Investigation of nonlinear dust-acoustic waves (DAWs) in a four-component unmagnetized electron–ion dusty plasma composed of fluid dust species, the Maxwellian positive, and negative ions with $q$ -nonextensive electrons is performed. The dust charge variation is considered for both adiabatic and nonadiabatic cases. For the nonadiabatic case, the Korteweg-de Vries–Burgers (KdV-B) equation is obtained utilizing the reductive perturbation technique (RPT). Implementing the traveling wave transfiguration, the KdV-B equation is transformed into a planar dynamical system (PDS). Implementing the phase plane theory of PDS, qualitative phase portrait profiles of a stable spiral for the corresponding system are presented. The effects of relevant physical parameters are shown on the DAW behavior. Dynamical features, such as quasiperiodic and chaotic motions of the system, are presented via a numerical investigation in the presence of an extraneous periodic force. Sensitivity analysis and the box-counting dimensions for chaotic profiles are presented. [ABSTRACT FROM AUTHOR]

Dynamical behavior of the optical traveling pulses for the resonant nonlinear Schrödinger (RNS) equation with external periodic force is studied. Using a complex transformation we obtain an unperturbed dynamical system for the RNS equation. Existence of periodic optical pulses, solitary optical pulses of dark and bright types, breaking optical pulses is dispensed using phase plane analysis of the unperturbed dynamical system. Introducing an external perturbation to the unperturbed dynamical system, quasiperiodicity and chaotic features of the nonlinear optical pulses for the perturbed dynamical system are studied by varying the resonance parameter (c) with special values of other system parameters through different computational tools, like time series plot, phase plot, sensitivity plot, Lyapunov exponent, and Poincare section. The resonance parameter (c) acts as a control parameter on qualitative transition of the nonlinear optical pulses for the perturbed dynamical system from quasiperiodic motion to chaotic motion. [ABSTRACT FROM AUTHOR]

A border p of a string x is an enhanced cover of x if the number of positions of x that lie within some occurrence of p is the maximum among all borders of x. (String p is a border of x if p is both a proper prefix and a suffix of x.) In this paper, two more general notions based on enhanced covers are introduced: a k -approximate enhanced cover and a relaxed k -approximate enhanced cover, where a fixed maximum number of errors k under the Hamming distance is considered. The k -approximate enhanced cover of x is its border, and its k -approximate occurrences are also considered in the covered number of positions of x. The relaxed k -approximate enhanced cover of x is a factor of x and a k -approximate border of x. Algorithms that compute all the variations of k -approximate enhanced covers mentioned above are presented in this paper. [ABSTRACT FROM AUTHOR]

Exact method of analytical solution of flat, non-dispersive eigenstates in a class of quasi-one-dimensional structures is reported within the tight-binding framework. The states are localized over certain sublattice sites. One such finite size cluster of atomic sites is decoupled from the rest of the system by the special "non-permissible" vertex having zero amplitude. This immediately leads to the self-trapping of the incoming excitation. We work out an analytical scheme to discern the localizing character of the diffraction free dispersionless modes using real space renormalization group technique. Supportive numerical calculations of spectral profile and transport are demonstrated to substantiate the essence of compact localized states. Possible experimental scope regarding the photonic analogue of the tight-binding electronic case is also discussed elaborately. This eventually unfolds the concepts of slow light and the related re-entrant mode switching from the study of optical dispersion. [ABSTRACT FROM AUTHOR]

Highlights • Dynamics of a ring of three unidirectionally-coupled nonidentical logistic maps has been analyzed. • Both "soft" and "hard" transitions to quasiperiodicity have been observed. • Noise-induced transitions from equilibrium and from two-cycle to quasiperiodicity have been discovered. • Noise-induced structural stabilization of the quasiperiodic attractor has been demonstrated. Abstract Transitions from equilibrium to quasiperiodicity and from a two-cycle to a quasiperiodic regime are studied in a ring of unidirectionally-coupled nonidentical logistic maps. The former scenario is realized through a "soft" (Neimark–Sacker) bifurcation, while the latter through a "hard" (saddle-node) bifurcation. Special attention is paid on a noise-induced transition through "hard" bifurcation, where a phenomenon of structural stabilization of the quasiperiodic system near the bifurcation point is observed and analyzed in detail. [ABSTRACT FROM AUTHOR]

The nearest-neighbor Aubry–André quasiperiodic localization model is generalized to include power-law translation-invariant hoppings T l ∝ t / l a or power-law Fourier coefficients W m ∝ w / m b in the quasiperiodic potential. The Aubry–André duality between T l and W m manifests when the Hamiltonian is written in the real-space basis and in the Fourier basis on a finite ring. The perturbative analysis in the amplitude t of the hoppings yields that the eigenstates remain power-law localized in real space for a > 1 and are critical for a c = 1 where they follow the strong multifractality linear spectrum, as in the equivalent model with random disorder. The perturbative analysis in the amplitude w of the quasiperiodic potential yields that the eigenstates remain delocalized in real space (power-law localized in Fourier space) for b > 1 and are critical for b c = 1 where they follow the weak multifractality Gaussian spectrum in real space (or strong multifractality linear spectrum in the Fourier basis). This critical case b c = 1 for the Fourier coefficients W m corresponds to a periodic function with discontinuities, instead of the cosinus function of the standard self-dual Aubry–André model. [ABSTRACT FROM AUTHOR]

Abstract Double-stranded quasiperiodic copper mean arrangement has been studied in respect of its electronic property and thermoelectric signature. The two-arm network is demonstrated by a tight-binding Hamiltonian. The eigenspectrum of such aperiodic mesh that does not convey translational invariance, is significantly dependent on the parameters of the Hamiltonian. It is observed that specific correlation between the parameters obtained from the commutation relation between the on-site energy and overlap integral matrices can eventually modify the spectral nature and generate absolutely continuous energy spectrum. This part is populated by atypical extended states that has a large localization length substantiated by the flow of the hopping integral under successive real space renormalization group method steps. This sounds delocalization of single particle energy states in such non-translationally invariant networks. Further this can be engineered at will by selective choice of the relative strengths of the parameters. This precise correlation has a crucial impact on the thermoelectric behavior. Anomalous nature of thermoelectric coefficient may inspire the experimentalists to frame tunable thermo-devices. Specific correlations can help us to tune the continuous band and determine the band position at will. Highlights • Electronic and thermoelectric properties of quasiperiodic ladder are reported. • Suitable parametric correlation induced exotic spectral feature is discussed. • Multichannel transport mechanism in support of spectral landscape is analyzed. • The tunability of transport and thermo-behavior may inspire the experimentalists. [ABSTRACT FROM AUTHOR]

Summary: The symmetry breaking of the loads of a matrix converter (MC) generated an imbalance that induced a quasiperiodic behavior at the input side of the MC. In addition, the load imbalance generated a negative‐sequence current at the input side. This paper adopts the time‐delayed feedback control (TDFC) to attenuate the quasiperiodic oscillation and the negative‐sequence component caused by the symmetry breaking. The results indicate that the attenuation of the quasiperiodic behavior directly leads to a decrease of the negative‐sequence current component. As a result, the TDFC improves the operation of the MC under unbalanced loads and part of the periodic symmetry is regained at the input side. The findings indicate that the quasiperiodicity and the amount of negative‐sequence current were strongly related. The proposed solution attenuated the quasiperiodic behavior while regaining part of the periodic symmetry. At the same time, the amount of negative‐sequence current decreased at the input side. [ABSTRACT FROM AUTHOR]

Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry-André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski-Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes. [ABSTRACT FROM AUTHOR]

This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with noncommensurate periods), in a way which does not require segmentation, training, object tracking, or 1-dimensional surrogate signals. Our methodology operates directly on video data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology) by translating the problem of finding recurrent dynamics in video data into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of our scores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished quantitatively end to end. [ABSTRACT FROM AUTHOR]

Elastic anti-plane shear waves can be refracted negatively when they are transmitted across an interface between a homogeneous substrate and a transverse periodic laminate. To achieve pure negative refraction, the frequency of the source should be lower than the upper limit of the second transition zone of the harmonic spectrum of the laminate. An effective way to control the location of transition zones is to consider a canonical configuration for the laminate, a concept that originates from the properties of quasicrystalline sequences among which the Fibonacci one is a particular case. We give a detailed account of the classification in three families of canonical configurations and the role of canonical frequency. We exploit the knowledge of the scaling factor of the self-similar structure of the layout of transition zones for laminates of this kind to provide a quantitative tool to predict the relevant frequencies to accomplish negative refraction. We also investigate how the change of other parameters of the elementary cell may affect the values of those frequencies. The obtained results show that the features of quasicrystalline sequences may be profitably exploited for the realisation of elastic metamaterials. • Canonical quasicrystalline laminates are defined and studied. • Their role in the problem of negative refraction is investigated. • Pass- and stop-band layouts are governed by scaling factors. [ABSTRACT FROM AUTHOR]

Motivated by the dynamics of a trimaran, an investigation of the dynamic behaviour of a double forcing parametrically excited system is carried out. Initially, we provide an outline of the stability regions, both numerically and analytically, for the undamped linear, extended version of the Mathieu equation. This paper then examines the anticipated form of response of our proposed nonlinear damped double forcing system, where periodic and quasiperiodic routes to chaos are graphically demonstrated and compared with the case of the single vertically-driven pendulum.

This research aims to characterize the nonlinear responses of a 3-layer laminated composite cantilever beam systemsubjected to evenly distributed dynamic excitations, with the employment of the Periodicity Ratio (PR) method. Periodic, quasiperiodic, chaotic behaviors especially the behaviors in between period and quasiperiod of the cantilever beam system are quantitatively described. A periodic-quasiperiodic-chaotic region diagram is developed to be used for identifying the global behaviors of the cantilever beam system with a large range of parameters. The research approach demonstrates that the PR method is effective and powerful in analyzing the complex dynamical characteristics of nonlinear cantilever beam systems, and the approach shows significances for industrial application and further nonlinear research in this area.