Discrete Systems Research Articles

Unconditionally convergent [formula omitted] splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equationwith the quasi-Riesz derivative.

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equationbased on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equationrepresents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Features of Network Flow Systems for Decision Making with Motivation Accounting

The present work investigates the problem of considering the influence of motivation in discrete network flow decision-making systems. Motivation is the primary factor that drives people to achieve their goals and plays a key role in decision-making systems. A method is proposed by which this influence is accounted for using the arc flow function amplification or attenuation coefficients. These operations are in the class of generalized network flows (flows with gains and losses). Methods to determine both the maximal and minimal value of the generalized network flow with consideration of the motivation are proposed A numerical example of a decision-making system based on a generalized network flow with motivation accounting is described, which confirms the obtained theoretical results.

Trajectory tracking control of discrete non-affine MIMO iterative systems with unknown models: a neural-network-based data-driven algorithm

Trajectory tracking control of discrete non-affine MIMO iterative systems with unknown models: a neural-network-based data-driven algorithm

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.

Model-based sparse optimal control of the hydrogen sulfide synthesis process for acidic wastewater sulfidation

Hydrogen sulfide is one of the commonly used materials of the gas-liquid sulfidation process for treating nonferrous metallurgical acidic wastewater. Using hydrogen and sulfur to synthesize hydrogen sulfide can obtain hydrogen sulfide with higher purity and at a lower cost. Due to the complex multiphase reaction mechanism of the hydrogen sulfide synthesis (HSS) process and production safety conditions, the manual operation requires a long time to ensure the continued stability of the system under high temperatures and high pressure. In this paper, a dynamic model and a novel sparse control strategy of the HSS process for acidic wastewater sulfidation are proposed to enable the HSS process operates continuously and safely. Firstly, a multivariate and multiphase mechanism model with multiple inputs and outputs of the HSS process is established. Subsequently, the novel sparse feedback matrix is designed under the constraint of the fixed l0-norm and the solvability of the eigenvalue determinant equation. The sparse feedback matrix solutions are optimized by pole placement and dynamic programming for continuous and discrete systems, respectively. Finally, the effectiveness of the HSS model is validated with industrial data. The superiority of the proposed model-based control strategy is demonstrated in the HSS system compared with conventional control methods.

Multi-index upper semicontinuity of pullback random attractors for fractional discrete FitzHugh-Nagumo systems with delays driven by Wong-Zakai approximations

In this paper, we consider the global well-posedness and multi-index stability of pullback random attractors for random fractional retarded lattice FitzHugh-Nagumo systems with nonlinear Wong-Zakai noise and non-autonomous forcing term. We use the idea of Caraballo et al. (2014) [2] to prove the global well-posedness. Based on the global well-posedness of the lattice FitzHugh-Nagumo system, we study the existence, uniqueness and upper semicontinuity of pullback random attractors Aϱδ={Aϱδ(τ,ω):τ∈R,ω∈Ω}. More precisely, we first utilize the method of tail-estimates of solution and Ascoli-Arzelà theorem to prove the existence and uniqueness of pullback random attractors, and then investigate the four types of upper semicontinuity of pullback random attractors: (1) The long time stability of pullback random attractors as the time parameter τ approaches negative infinity; (2) The upper semicontinuity of pullback random attractors as the step-length δ of the Wong-Zakai approximation tends to positive infinity; (3) The upper semicontinuity of pullback random attractors as the delay time ϱ→0; (4) The upper semicontinuity of pullback random attractors from non-autonomous to autonomous.

Yoneyama's 3/2 test for the asymptotic stability of delay dynamic equations

We consider the linear delay dynamic equation (*) x Δ ( t ) + p ( t ) x ( τ ( t ) ) = 0 for t ∈ [ t 0 , ∞ ) T . We enhanced the results in Wu and Zhou [2007. Stability for first order delay dynamic equations on time scales. Computers and Mathematics with Applications, 53(12), 1820–1831] by removing the boundedness criterion from the delay and graininess functions to cover the result in Zhang et al. [1999. Global attractivity of difference equations with variable delay. Dynamics of Continuous, Discrete and Impulsive Systems, 6(3), 307–317]. Furthermore, we have defined new functions that will assist us in proving our main results. We have presented some examples that illustrate the essence of our results in this work.

Mapping Petri Nets onto a Calculus of Context-Aware Ambients

Petri nets are a graphical notation for describing a class of discrete event dynamic systems whose behaviours are characterised by concurrency, synchronisation, mutual exclusion and conflict. They have been used over the years for the modelling of various distributed systems applications. With the advent of pervasive systems and the Internet of Things, the Calculus of Context-aware Ambients (CCA) has emerged as a suitable formal notation for analysing the behaviours of these systems. In this paper, we are interested in comparing the expressive power of Petri nets to that of CCA. That is, can the class of systems represented by Petri nets be modelled in CCA? To answer this question, an algorithm is proposed that maps any Petri net onto a CCA process. We prove that a Petri net and its corresponding CCA process are behavioural equivalent. It follows that CCA is at least as expressive as Petri nets, i.e., any system that can be specified in Petri nets can also be specified in CCA. Moreover, tools developed for CCA can also be used to analyse the behaviours of Petri nets.

Measuring Residual Stresses with Crack Compliance Methods: An Ill-Posed Inverse Problem with a Closed-Form Kernel

By means of relaxation methods, residual stresses can be obtained by introducing a progressive cut or a hole in a specimen and by measuring and elaborating the strains or displacements that are consequently produced. If the cut can be considered a controlled crack-like defect, by leveraging Bueckner’s superposition principle, the relaxed strains can be modeled through a weighted integral of the residual stress relieved by the cut. To evaluate residual stresses, an integral equation must be solved. From a practical point of view, the solution is usually based on a discretization technique that transforms the integral equation into a linear system of algebraic equations, whose solutions can be easily obtained, at least from a computational point of view. However, the linear system is often significantly ill-conditioned. In this paper, it is shown that its ill-conditioning is actually a consequence of a much deeper property of the underlying integral equation, which is reflected also in the discretized setting. In fact, the original problem is ill-posed. The ill-posedness is anything but a mathematical sophistry; indeed, it profoundly affects the properties of the discretized system too. In particular, it induces the so-called bias–variance tradeoff, a property that affects many experimental procedures, in which the analyst is forced to introduce some bias in order to obtain a solution that is not overwhelmed by measurement noise. In turn, unless it is backed up by sound and reasonable physical assumptions on some properties of the solution, the introduced bias is potentially infinite and impairs every uncertainty quantification technique. To support these topics, an illustrative numerical example using the crack compliance (also known as slitting) method is presented. The availability of the Linear Elastic Fracture Mechanics Weight Function for the problem allows for a completely analytical formulation of the original integral equation by which bias due to the numerical approximation of the physical model is prevented.

Diagnosability and attack detection for discrete event systems under sensor attacks

Diagnosability and attack detection for discrete event systems under sensor attacks

Chaos synchronization of two coupled map lattice systems using safe reinforcement learning

When synchronizing two continuous typical chaotic systems, the state variables of the response system are usually bounded and satisfy the Lipschitz condition. This permits a universal synchronization method. In the synchronization of two high-dimensional discrete chaotic systems such as the coupled map lattice systems, the state variables of the response system often diverge, which makes it difficult to seek a universal synchronization method. To overcome this difficulty, bounded and hard constraints, which correspond to the concept of safety level III in reinforcement learning terms, on the state variables must be imposed, when the discrete response system is perturbed. We propose a universal method for synchronizing two discrete systems based on safe reinforcement learning (RL). In this method, the RL agent’s policy is used to reach the goal of synchronization and a safety layer added directly on top of the policy is used to satisfy safety level III. The safety layer consists of a one-step predictor for the perturbed response system and an action correction formulation. The one-step predictor, based on a next generation reservoir computing, is used to identify whether the next state of the perturbed system is within the chaotic domain, and if not, the action correction formula is activated to modify the corresponding perturbing force component to zero. In this way, the state of the perturbed system will remain in the bounded chaotic domain. We demonstrate that the proposed method succeeds in synchronization without divergence through a numerical example with two coupled map lattice systems, and the average synchronization time is 12.66 iteration steps over 1000 different initial conditions. The method works even when parameter mismatches and disturbances exist in the system. We compare the performance in both cases with and without the safety layer and find that successful synchronization is only possible in the former case, emphasizing the significance of the safety layer. The performance of the algorithm with optimal hyper-parameters obtained by Bayesian optimization is robust and stable.

What is finiteness in Dagaare?

Abstract Although the notion of finiteness has been much debated from different theoretical perspectives, little is known beyond Indo-European languages. The present study examines finiteness in the Lobr dialect of Dagaare (Niger-Congo: Mabia) from a systemic functional typological point of view. The study contributes to debates on whether finiteness is a discrete system or a cline. The study first distinguishes between semantic and grammatical finiteness. The analysis then show that Dagaare has a grammaticalised binary system of finiteness in the verbal group, encoded by four kinds of finiteness markers: (i) tense-mood-polarity, (ii) modality, (iii) habitual tense, and (iv) remoteness particles. Also, the Dagaare verbal group embodies disassociated systems of tense, comprising futurity, habituality, and remoteness and choices from each of these systems result in multiple Finite elements in the verbal group. On the other hand, clausal finiteness in Dagaare is a semantic cline that depends on the system of freedom. Specifically, a clause is finite or less finite depending on whether it combines a finite verbal group with clause final negotiation or information focus. The article provides finer criteria for distinguishing between semantic and grammatical finiteness that can be used for the description and typology of finiteness across languages.

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Self-Assembly of Discrete Multi-Chromophoric Systems.

Self-assembly of chromophoric systems is a prerequisite to create well-ordered, processable nanomaterials with multiple functionalities. In the past two decades, the field of functional organic materials has primarily focused on systems featuring only one type of dye/π-conjugated unit. Consequently, many reports with mechanistic insights on the self-assembly of the dyes featuring different molecular packing have been reported. Subsequently, we have witnessed several attempts to organize the multi-chromophoric systems in solution and solid-state via different approaches using self-assembly as a tool. Incorporation of more than one dye is important in creating materials with tuneable optoelectronic properties. Consequently, self-assembly of more than one chromophoric systems have been investigated to some extent. This review aims to discuss the self-assembled materials derived from discrete π-conjugated systems comprising more than one dye units connected through covalent bonding (multi-chromophoric systems). Molecular design of various multi-chromophoric systems leading to the formation of crystals, liquid crystals and supramolecular polymers have been correlated with corresponding properties. We envisage that classification of self-assembled multi-chromophoric systems, with a note on tuneable optoelectronic properties, can provide a deeper understanding on the molecular design strategies, which is important in the fabrication of functional organic materials with optimum performances.

Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems

Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems

Lagrangian multiform structure of discrete and semi-discrete KP systems

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.

Symmetric Pseudo-Multi-Scroll Attractor and Its Application in Mobile Robot Path Planning

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing.