Discrete Systems Research Articles

Thermalization dynamics in photonic lattices of different geometries

The statistical mechanical behavior of weakly nonlinear multimoded optical settings has been attracting increased interest over the last few years. The main purpose of this work is to numerically investigate the main factors that affect the thermalization process in photonic lattices. In particular, we find that lattices with identically selected properties (such as temperature, coupling coefficient, lattice size, and excitation conditions) can exhibit very different thermalization dynamics and, thus, thermalization distances. Our investigation is focused on two different two-dimensional lattices: the honeycomb lattice and the triangular lattice. Our numerical results show that, independently of the excitation conditions, the honeycomb lattice always thermalizes faster than the triangular lattice. We mainly explain this behavior by the quasilinear spectrum that promotes wave-mixing in the honeycomb lattice in comparison to the power-like spectrum of the triangular lattice. In addition, we investigate the combined effects of temperature as well as the sign and magnitude of the nonlinearity. Switching either the sign of the Kerr nonlinear coefficient or the sign of the temperature can lead to significant differences in the thermalization dynamics, a phenomenon that can be physically explained in terms of wave instabilities. Larger absolute values of the temperature |T| result in more uniform distributions for the power occupation numbers and faster thermalization speeds. Finally, as expected, increasing the magnitude of the nonlinearity results in accelerated thermalization. Our findings provide valuable insights into optical thermalization in discrete systems, where experimental realization may bring about new possibilities for light manipulation and applications.

Identification and tracing of near-source petroleum transport system based on reservoir bitumen index (RBI) method: A case study of the lower cambrian in the Tazhong-Bachu area, Tarim Basin, China

The exploration of deep layers has become increasingly important in the global oil and gas industry. The Tazhong-Bachu area of the Tarim Basin is a pioneering target for deep petroleum exploration in China, but only Wells Zhongshen1 and Zhongshen5 have found industrial oil flow in the Cambrian. Noteworthily, the occurrence of reservoir bitumen in the Lower Cambrian coring interval in many wells indicates that large-scale hydrocarbon migration had occurred here in geological history. Effective identification of reservoir bitumen in the Cambrian dolomite reservoirs is crucial to understanding hydrocarbons' distribution and migration. In this study, we adopt the Reservoir Bitumen Index (RBI) method to deduce a quantitative calculation formula for reservoir bitumen, and classify the transport system into four types based on differences in hydrocarbon transport behavior and characteristics. The results show that the deep carbonate low permeability-tight reservoirs of the Lower Cambrian in the Tazhong-Bachu area generally develop reservoir bitumen, most likely derived from underlying Precambrian source rocks. Therefore, the Lower Cambrian carbonate reservoir is considered a near-source discrete petroleum transport system, providing great potential for further oil and gas exploration in the Lower Paleozoic in the Tazhong-Bachu area.

Supervisory control under delayed observations of events and states

In discrete event systems, it is often convenient and possible to observe directly whether or not the system is in a subset of the state space, typically after some delay, even if some event occurrences leading to the current state were not observed. We model supervisory control with the delayed observations of events and states and investigate the existence of a supervisor to obtain a given desired language accordingly. An existence verifier with polynomial run time is presented.

An improved multiple discrete time system for operational modal analysis of a slanted rotational blade under natural excitation via a long-range tracking continuously scanning laser Doppler vibrometer system

A long-range tracking continuously scanning laser Doppler vibrometer (LTCSLDV) system is able to continuously track the speed and simultaneously sweep the surface of a distanced rotational blade with predicted paths. During the long-range continuous scanning, laser beams including raw responses with modal parameter information are transmitted from the measured distanced rotational blade to the laser Doppler vibrometer (LDV). However, classical signal processing methods for raw responses obtained from a LTCSLDV measurement, which are the polynomial and demodulation method, cannot implement standard modal estimation. Its reason is that continuous scanning responses includes density virtual measuring points and one of its results is that some significant modal parameter, especially damping ratio, cannot be estimated directly. This work provides an improved multiple discrete time system (MDTS) for LTCSLDV measurements that is capable of extracting the responses of multiple virtual measuring points on the virtual scanning path with equally intervals as if these responses are measured at fixed pseudo-measurement points. Therefore, extensively complex signal processing can be implemented on the processed responses through program or commercial software only if the amount of the data of them is sufficient for modal analysis. The measurements were performed on a slanted rotational blade (SRB) in real operational condition with the considerations of the pitch angle of the SRB and the tilt angle of the direction of laser beam emitted from a LDV. The predicted tracking and scanning path of the laser spot in long-range measurements is realized through visual identification with a high-speed color camera.

ℒ2 gain tracking control of linear completely unknown discrete‐time networked control systems with dropout

AbstractWe introduce an online, model‐free algorithm to address the gain optimal tracking problems in discrete linear networked control systems. The algorithm is specifically proposed to handle stochastic information dropout in the feedback loop. Our goal is to design a control law that achieves the system output reference tracking while attenuating the effect of the disturbance input. We first construct an augmented system consisting of the original system and the command generator system. Then, the performance index is expressed in a quadratic form taking into account packet loss. Next, we obtain the optimal solution by solving the dropout generalized algebraic riccati equation (GARE). Finally, a ‐learning algorithm is utilized to estimate the control and disturbance feedback gains of the system, using only measurement data with unknown system dynamics in the presence of dropout. Two algorithms are tested on a numerical example to demonstrate the validity and effectiveness of the proposed methodology.

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

On a sharp estimate of overlapping Schwarz methods in H(curl; Ω) and H(div; Ω)

Abstract The previously proved upper bound is $C\big(1+\frac{H^{2}}{\delta ^{2}}\big)$ for the condition number of the discrete system arising from the classical two-level overlapping Schwarz method for symmetric positive definite problems in $\boldsymbol{H}(\textbf{curl}; \varOmega )$ and $\boldsymbol{H}(\textrm{div}; \varOmega )$, where $H$ and $\delta $ are the sizes of subdomains and overlap, respectively. But all numerical results indicate that a sharper bound is $C\big (1+ \frac{H}{\delta } \big )$. In this work, we prove that the condition number is indeed bounded above by $C\big(1+\frac{H}{\delta }\big)$.

Master–slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this paper, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition.

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale ω between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value ω=ω_{c} when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ω_{c} in terms of the coupling parameters of the system. We show that ω_{c} is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ω_{c} from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ω^{*} of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix A and compressibility matrix K. Our exact calculations show that Onsager-type current symmetry relation AK=KA^{T} can be satisfied if and only if ω^{*}=ω_{c}. Our coarse-grained model calculations can be easily generalized to other coupled systems.

A Three-Grid High-Order Immersed Finite Element Method for the Analysis of CAD Models

The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains described by complex CAD models and discretised with tens of millions of degrees of freedom. The solution field is discretised using linear and quadratic Lagrange basis functions.

Frequency-Histogram Coarse Graining in Elementary and 2-Dimensional Cellular Automata

Cellular automata and other discrete dynamical systems have long been studied as models of emergent complexity. Recently, neural cellular automata have been proposed as models to investigate the emerge of a more general artificial intelligence, thanks to their propensity to support properties such as self-organization, emergence, and open-endedness. However, understanding emergent complexity in large scale systems is an open challenge. How can the important computations leading to emergent complex structures and behaviors be identified? In this work, we systematically investigate a form of dimensionality reduction for 1-dimensional and 2-dimensional cellular automata based on coarse-graining of macrostates into smaller blocks. We discuss selected examples and provide the entire exploration of coarse graining with different filtering levels in the appendix (available also digitally at this link: https://s4nyam.github.io/eca88/. We argue that being able to capture emergent complexity in AI systems may pave the way to open-ended evolution, a plausible path to reach artificial general intelligence.

Understanding the influence of confounding factors in myoelectric control for discrete gesture recognition

Discrete myoelectric control-based gesture recognition has recently gained interest as a possible input modality for many emerging ubiquitous computing applications. Unlike the continuous control commonly employed in powered prostheses, discrete systems seek to recognize the dynamic sequences associated with gestures to generate event-based inputs. More akin to those used in general-purpose human-computer interaction, these could include, for example, a flick of the wrist to dismiss a phone call or a double tap of the index finger and thumb to silence an alarm. Moelectric control systems have been shown to achieve near-perfect classification accuracy, but in highly constrained offline settings. Real-world, online systems are subject to ‘confounding factors’ (i.e. factors that hinder the real-world robustness of myoelectric control that are not accounted for during typical offline analyses), which inevitably degrade system performance, limiting their practical use. Although these factors have been widely studied in continuous prosthesis control, there has been little exploration of their impacts on discrete myoelectric control systems for emerging applications and use cases. Correspondingly, this work examines, for the first time, three confounding factors and their effect on the robustness of discrete myoelectric control: (1) limb position variability, (2) cross-day use, and a newly identified confound faced by discrete systems (3) gesture elicitation speed. Results from four different discrete myoelectric control architectures: (1) Majority Vote LDA, (2) Dynamic Time Warping, (3) an LSTM network trained with Cross Entropy, and (4) an LSTM network trained with Contrastive Learning, show that classification accuracy is significantly degraded ( p< 0.05) as a result of each of these confounds. This work establishes that confounding factors are a critical barrier that must be addressed to enable the real-world adoption of discrete myoelectric control for robust and reliable gesture recognition.

Employing discrete global grid systems for reproducible data obfuscation

Archaeological heritage worldwide is threatened through deliberate destruction in particular site looting making the location of archaeological sites potentially sensitive data. At the same time, public information about site locations are important for heritage management, as well as agricultural and urban development. Finding a balance between revealing detailed site locations and not providing data at all is a difficult task. Here we provide an approach to obfuscate archaeological site location data facilitated through a Discrete Global Grid System. We then apply the new obfuscation method to the global p3k14c data set. Veiling the locations of heritage sites with a Discrete Global Grid System allows tiered accuracy access for different stakeholders tailored to their respective needs as well as legal constraints and requirements of administrations. Discrete Global Grid System based obfuscation is globally scalable, consistent, reproducible, and can address the current heterogeneity of obfuscation methods.

Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant argument

This study marks the first exploration of fuzzy linear fractional differential equations with a piecewise constant argument (FLFDEs-PCA), incorporating the concept of Caputo’s type gH-differentiability with the order α ∈ (0, 1]. Such problems are noteworthy as they represent hybrid systems, blending the characteristics of continuous and discrete dynamical systems and integrating aspects from both differential and difference equations. The primary objective of this research is to establish a standardized framework for deriving explicit solution formulas for FLFDEs-PCA under various scenarios. Additionally, illustrative examples are provided to demonstrate the practical implications of our theoretical findings.

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.

Modeling the dynamics of the Hepatitis B virus via a variable-order discrete system

We investigate the dynamics of the hepatitis B virus by integrating variable-order calculus and discrete analysis. Specifically, we utilize the Caputo variable-order difference operator in this study. To establish the existence and uniqueness results of the model, we employ a fixed-point technique. Furthermore, we prove that the model exhibits bounded and positive solutions. Additionally, we explore the local stability of the proposed model by determining the basic reproduction number. Finally, we present several numerical simulations to illustrate the richness of our results.

Highly accurate wavelet solution for the two-dimensional Bratu's problem

A modified wavelet interpolation Galerkin method (WIGM) with improved flexibility of nodal distribution is formulated and utilized to solve the Bratu's problem. Such a method treats the nonlinear term as an independent function and directly approximates it through the proposed wavelet interpolation, resulting in a low-cost discretization of the Bratu's equation with exponential nonlinearity. An efficient implementation of Newton's method to find the lower and upper branches of solution as well as the turning point is then presented. Numerical results demonstrate that the present WIGM can accurately estimate both two solution branches and turning point for the Bratu's problem. Compared with many other existing methods, the proposed WIGM also offers marked superiority in terms of accuracy when using the same node distribution. Moreover, a convergent solution to the resulting nonlinear discrete system obtained by the WIGM can be observed in a few iterations even when the initial guess significantly deviates from the exact solution. Finally, a highly accurate approximation of both two-branched solutions and the turning point for the two-dimensional Bratu's problem is obtained by the proposed WIGM.

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0$$\\end{document}, stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when R0=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0=1$$\\end{document}. It is concluded that the endemic equilibrium is globally asymptotically stable when R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}_0>1$$\\end{document}, by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

Discrete Dynamic System Modeling for Simulated Moving Bed Processes

Although the simulated moving bed (SMB) process boasts advantages such as high productivity and low consumption, the cost of obtaining optimized parameters through practical experiments to control the separation process can be enormous due to its complex nonlinear characteristics. Consequently, the successful transformation of the SMB separation process into a mathematical dynamic model for computer simulation would greatly reduce the research costs associated with experimental studies. In this study, the Crank–Nicolson method was employed to discretize and dynamize the SMB process, enabling the simulation of processes under both linear and Langmuir isotherms. The results of the simulation experiments demonstrated the feasibility and high efficiency of this approach, thereby establishing a solid foundation for further advancements in online control strategies.

A general modeling scheme for spatiotemporal DGGS with emphasis on encoding and operating multiscale time grids

AbstractOne of the basic scientific problems concerning geographic information science is how to rapidly organize, query, and compute spatiotemporal big data. The spatiotemporal discrete global grid system (DGGS) provides a homogenized discrete structure for processing multiscale and multitype spatiotemporal data. To date, most research in spatiotemporal DGGS has focused on spatial discretization while neglecting temporal discretization. Here, we propose a general modeling scheme for spatiotemporal DGGS with emphasis on encoding and operating multiscale time grids. We subdivide continuous time into multiscale temporal grids, which are then encoded as integers. Moreover, we designed integer code operations, including hierarchical traversal, neighborhood finding, and temporal relationship calculations. Compared to the multiscale time segment integer coding (MTSIC) approach, the proposed method resulted in 22% higher encoding efficiency, 10.92 times faster decoding, 2.81 times better parent code finding efficiency, 41% improved efficiency, 100% accuracy in finding children codes (compared to less than 100% with MTSIC), and a 62% enhancement in temporal relationship calculation efficiency. The application of querying spatiotemporal trajectory data validates the feasibility and practicality of substituting conventional string‐based time and floating‐point location coordinates with spatiotemporal integer codes to query data. The time encoding and operation methods proposed here indicate high efficiency, superior accuracy, and broad application prospects.