Aperiodic Systems Research Articles

Continuous approximation of linear impulsive systems and a new form of robust stability

The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, producing what is known as an impulse extension equation. Theoretical properties related to the existence of the time-scale tolerance are given for periodic systems, as are algorithms to compute them. Some methods are presented for general, aperiodic systems. Additionally, sufficient conditions for the convergence of solutions of impulse extension equations to the solutions of their associated impulsive differential equation are proven. Counterexamples are provided.

Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems

Sufficient conditions for the design of a simple class of interval observers for linear impulsive systems subject to minimum and range dwell-time constraints are obtained and formulated in terms of infinite-dimensional linear programs. The proposed approach is fully constructive in the sense that suitable observer gains can be extracted from the solution of the optimization problems and is flexible enough to be extended to include performance constraints and tackle parametric uncertainties. In order to be solvable, the infinite-dimensional linear programs are relaxed using a method based on sum of squares which is known to be asymptotically exact in the present case. Three examples are given for illustration: the first one pertains on the interval observation of an impulsive system under a minimum dwell-time constraint, the second one is about the interval observation of an aperiodic sampled-data system and the last one is about the interval observation of a linear switched system.

Phononic Localization in Locally Resonant Ternary Thue-Morse and Rudin-Shapiro Structures

The wave localization properties in locally resonant ternary phononic aperiodic crystals based on Thue-Morse and Rudin-Shapiro sequences, are theoretically investigated by the transfer matrix method. The various factors affecting localization characteristics are discussed. Numerical results show that the effect of material properties and the geometric structure parameters on the number and size of band gaps is obvious. Moreover, compared with the corresponding binary aperiodic structures without local resonances, the localization factors become larger in the locally resonant ternary structures. And wider and lower-frequency band gaps and localized resonant flat bands appear in the locally resonant aperiodic system. Compared with the locally resonant Thue-Morse system, the localization factor becomes larger and the band splitting phenomenon is less obvious in the locally resonant Rudin-Shapiro system, which is just the opposite to that in the aperiodic system without local resonance. This study is useful in filter, waveguide, and so on.

The order-disorder evolution in quasicrystals through phason flips

Abstract The problem of incorporating phason flips in the structural investigation of aperiodic systems is still an open question and a challenge in crystallography. Phasons are understood both as atomic fluctuation and positional disorder of the quasicrystalline lattice. Popular correction to diffraction peaks' intensities takes the form of generalized Debye-Waller factor, assuming Gaussian distribution of fluctuations in the perpendicular space of higher-dimensional periodic lattice. Although proven to work in case of random tiling types of structures recent evidence indicates improper handling of peaks with high perpendicular space scattering vector whenever structure is far from random tiling regime. We introduce the concept of a series expansion of the characteristic function of the statistical distribution to properly correct the peaks' intensities with respect to phasonic fluctuations. Calculations are performed upon Penrose tiling. Such approximation of the structure factor works correctly even in cases for which the Debye-Waller correction fails. Even more we investigate transition to random tiling through phason flips by means of the statistical approach which results in interesting scaling properties (ordered → disordered → random → amorphous structure).

Dwell‐time‐dependent stability results for impulsive systems

This study is concerned with the dwell-time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov-like functional approach is established to study the stability for impulsive systems. The Lyapunov-like functional is time-varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov-like functional is constructed by introducing the integral of state together with the cross-terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov-like functional approach, new dwell-time-dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples.

Quo Vadis Quasicrystals?

This Special Issue aims at gaining a deeper understanding on the relationship between the underlying structural order and the resulting physical properties in aperiodic systems, including quasicrystalline and related complex metallic alloys, photonic quasicrystals, and other structures exhibiting long-range aperiodic order. This Special Issue contains 12 papers which highlight recent developments in quasiperiodic crystal structure, photonic quasicrystals and related optical devices, the intrinsic electrical, thermal, and mechanical properties of icosahedral and decagonal metallic alloys, and the nature of chemical bonding in intermetallic compounds, from a multidisciplinary perspective. In light of the results presented in the contributions collected in this Special Issue, we can confidently expect that new insights into the interdisciplinary science of quasicrystals will be gained in the years to come, providing a sharper picture of their structures and related physical properties, and spurring further progress in practical issues related to both materials engineering science and nanotechnology.

Finite-temperature calculations of the Compton profile of Be, Li, and Si

High resolution inelastic x-ray scattering experiments are widely used to study the electronic and chemical properties of materials under a range of conditions, from ambient temperature to the warm dense matter regime. We use the real-space multiple scattering (RSMS) Green's function formalism coupled with density functional theory molecular dynamics (DFT-MD) to study thermal effects on the Compton profile (CP) of disordered systems. The RSMS method is advantageous for calculations of highly disordered, aperiodic systems because it places no restriction on symmetry. As a test, we apply our approach to thermally disordered Be, Li, and Si in both liquid and solid phases. We find good agreement with experimental and other theoretical results, showing that the real-space multiple scattering approach coupled with DFT-MD is an efficient and reliable method for calculating the CP of disordered systems at finite temperatures.

Asynchronous sampled-data approach for event-triggered systems

ABSTRACTWhile aperiodically triggered network control systems save a considerable amount of communication bandwidth, they also pose challenges such as coupling between control and event-condition design, optimisation of the available resources such as control, communication and computation power, and time-delays due to computation and communication network. With this motivation, the paper presents separate designs of control and event-triggering mechanism, thus simplifying the overall analysis, asynchronous linear quadratic Gaussian controller which tackles delays and aperiodic nature of transmissions, and a novel event mechanism which compares the cost of the aperiodic system against a reference periodic implementation. The proposed scheme is simulated on a linearised wind turbine model for pitch angle control and the results show significant improvement against the periodic counterpart.

Aperiodic Weak Topological Superconductors.

Weak topological phases are usually described in terms of protection by the lattice translation symmetry. Their characterization explicitly relies on periodicity since weak invariants are expressed in terms of the momentum-space torus. We prove the compatibility of weak topological superconductors with aperiodic systems, such as quasicrystals. We go beyond usual descriptions of weak topological phases and introduce a novel, real-space formulation of the weak invariant, based on the Clifford pseudospectrum. A nontrivial value of this index implies a nontrivial bulk phase, which is robust against disorder and hosts localized zero-energy modes at the edge. Our recipe for determining the weak invariant is directly applicable to any finite-sized system, including disordered lattice models. This direct method enables a quantitative analysis of the level of disorder the topological protection can withstand.

Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples

This article presents, in an illustrative fashion, a first step towards an extension of the spectral theory of constant length substitutions. Our starting point is the general observation that the symbolic picture (as defined by the substitution rule) and its geometric counterpart with natural prototile sizes (as defined by the induced inflation rule) may differ considerably. On the geometric side, an aperiodic inflation system possesses a set of exact renormalisation relations for its pair correlation coefficients. Here, we derive these relations for some paradigmatic examples and infer various spectral consequences. In particular, we consider the Fibonacci chain, revisit the Thue–Morse and the Rudin–Shapiro system, and finally analyse a twisted extension of the silver mean chain with mixed singular spectrum.

Universal Systems for Entropy Intervals

A topological system is universal for a class of ergodic measure-theoretic systems if its simplex of invariant measures contains, up to an isomorphism, all elements of this class and no elements from outside the class. We construct universal systems for classes given by the combination of three properties: measure-theoretic entropy belonging to a nondegenerate interval of the extended nonnegative real halfline, invertibility and aperiodicity. For classes consisting of aperiodic systems the universal system can be made minimal.

Stability analysis and stabilization of aperiodic sampled-data systems based on a switched system approach

This paper investigates the stability analysis and controller design problems for aperiodic sampled-data control systems. The time-varying sampling period is assumed to take only a finite number of values, and the considered system is modeled as a discrete-time switched system with both stable and unstable subsystems. A sufficient condition is derived for the system to be exponentially stable, and a state feedback stabilizing controller is designed such that the system is exponentially stable and achieves an optimal guaranteed cost performance. The obtained exponential decay rate and guaranteed cost performance level critically depend on both the lengths and activation frequencies of the varying sampling periods, and it is revealed that nonuniformly sampled control systems can remain to be stable even in the appearance of a very large sampling period, provided that the activation frequency of the large sampling period is small enough. Finally, an illustrative example is given to show the effectiveness of the proposed approach.

Two-level design for aperiodic networked control systems

Two critical issues in networked control systems are coupling of control and communication, and energy economy, especially for battery powered wireless sensor nodes. In this regard, a two-level design procedure for multiple loops with H∞ based self-triggered control applied over the modified IEEE 802.15.4 wireless protocol is presented. Control and communication are decoupled by introducing an upper-bound on the transmission rate, and optimal usage of communication bandwidth and energy is guaranteed by modifying the wireless protocol. The presented priority-based scheduling algorithm can accommodate more systems as compared with the number of available transmission slots, while being efficient in terms of computational load and energy consumption. Simulation results show a significant reduction in energy expenditure in terms of decrease in duty cycle as compared with the periodic implementation.

Embedding topological dynamical systems with periodic points in cubical shifts

According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\text{shift})$ if both its mean dimension and periodic dimension are strictly bounded by $d/2$. We verify the conjecture for the class of systems admitting a finite-dimensional non-wandering set and a closed set of periodic points. This class of systems is closely related to systems arising in physics. In particular, we prove an embedding theorem for systems associated with the two-dimensional Navier–Stokes equations of fluid mechanics. The main tool in the proof of the embedding result is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin lemma. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that, for systems with the marker property, vanishing mean dimension is equivalent to the small boundary property.

Stability of aperiodic sampled-data systems — An augmented looped Lyapunov functional with wirtinger inequality approach

This paper is concerned with stability analysis of sampled-data systems with non-uniform sampling patterns. The stability problem is solved based on an augmented looped Lyapunov functional with Wirtinger inequality approach. The effectiveness of this approach depends on proper selecting looped Lyapunov functional parts and less conservative inequality techniques. The method developed here is an extension of previous results by using novel augmented functional and Wirtinger inequality. Numerical examples are given to illustrate the result.

Ballistic conduction in macroscopic non‐periodic lattices

In this paper, the dc electronic transport at zero temperature in aperiodic systems with macroscopic length is studied by using a real‐space renormalization plus convolution method developed for the Kubo–Greenwood formula within the tight‐binding formalism. We analytically prove the existence of transparent states in several generalized Fibonacci lattices, as well as in segmented linear chains, where they always appear if the number of bonds in each segment is even, regardless the ordering of segments. For two‐dimensional square‐lattice tapes with a periodic or non‐periodic Fano‐impurity plane, we found a novel ballistic transport state in the dc conductance spectra.

Generation of extended states in diluted transmission lines with distribution of inductances according to Galois sequences: Hamiltonian map approach

We study the localization properties of diluted direct transmission lines, when we distribute two values of inductances LA and LB, according to the aperiodic Galois sequence. When we dilute the aperiodic Galois system with (d−1) inductances with constant L0 value, we find d sub-bands and (d−1) gaps; here d is the period of the distribution of the Galois sequence in the diluted system. Under the condition L0≈(LA,LB), we find a set of extended states for finite Nd system size, which disappears when Nd→∞. For the case L0⪢(LA,LB), using the scaling behavior of the averaged participation number 〈D(ω)〉 and the scaling behavior of the averaged normalized participation number 〈ξ(ω)〉, we demonstrate the existence of extended states in the thermodynamic limit.

Distributed Sampled-Data <formula formulatype="inline"><tex Notation="TeX">${H_\infty }$</tex> </formula> Filtering for Sensor Networks With Nonuniform Sampling Periods

This paper presents a switched system approach to solving the distributed sampled-data <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mbi{H_\infty }$</tex> </formula> filtering problem for sensor networks with nonuniform sampling periods. The sensor network is considered to be a peer-to-peer network without an estimation center. The measurements are sampled with nonuniform sampling periods, and each sensor in the network collects the sampled measurements only from its neighbors and runs a distributed <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mbi{H_\infty }$</tex> </formula> filtering algorithm to generate estimates. A stochastic switched system model is proposed to describe the aperiodic sampled-data filtering system with random packet losses. A sufficient existence condition for the distributed <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mbi{H_\infty }$</tex> </formula> filters is derived by using the average dwell time method, and it is shown that the obtained condition critically depends on the sampling periods and the packet loss probabilities. The design of the filters is accomplished by solving a convex optimization problem, and the designed filters guarantee that the filtering system is mean-square exponentially stable and all the filtering errors satisfy an average <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mbi{H_\infty }$</tex> </formula> noise attenuation level. An illustrative example is finally given to show the effectiveness of the proposed results.

Normal form and Nekhoroshev stability for nearly integrable hamiltonian systems with unconditionally slow aperiodic time dependence

The aim of this paper is to extend the results of Giorgilli and Zehnder for aperiodic time dependent systems to a case of general nearly-integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent on the size of the perturbation.

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years. In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space.