Discrete Dynamical Systems Research Articles

An overview of numerical methods for discrete dynamical systems with frictionless unilateral constraints: mathematical issues and convergence results

An overview of numerical methods for discrete dynamical systems with frictionless unilateral constraints: mathematical issues and convergence results

Synthesis of an Optimal Control for Linear Stationary Discrete Dynamical Systems

Synthesis of an Optimal Control for Linear Stationary Discrete Dynamical Systems

Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) (H,f1,∞) and its induced set-valued discrete dynamical systems (K(H),f¯1,∞). Specifically, it explores the chaotic properties of these systems. The main finding is that f1,∞ is Devaney chaotic if and only if f¯1,∞ is Devaney chaotic in we-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.

Models of Cell Processes are Far from the Edge of Chaos.

Complex living systems are thought to exist at the "edge of chaos" separating the ordered dynamics of robust function from the disordered dynamics of rapid environmental adaptation. Here, a deeper inspection of 72 experimentally supported discrete dynamical models of cell processes reveals previously unobserved order on long time scales, suggesting greater rigidity in these systems than was previously conjectured. We find that propagation of internal perturbations is transient in most cases, and that even when large perturbation cascades persist, their phenotypic effects are often minimal. Moreover, we find evidence that stochasticity and desynchronization can lead to increased recovery from regulatory perturbation cascades. Our analysis relies on new measures that quantify the tendency of perturbations to spread through a discrete dynamical system. Computing these measures was not feasible using current methodology; thus, we developed a multipurpose CUDA-based simulation tool, which we have made available as the open-source Python library cubewalkers. Based on novel measures and simulations, our results suggest that-contrary to current theory-cell processes are ordered and far from the edge of chaos.

Stochastic Process Leading to Catalan Number Recurrence

Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of {1,2,…,N}, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system.

A Study on Some Properties for Weak Stability of Non-Autonomous Discrete Dynamical Systems

The interest for non-autonomous discrete dynamical systems has been increasing in recent years, because they are adequate to tell of real activities. For examples, when the mapping is disturbed in each iteration because of external factors, or model of some phenomena in physics, biology and economy, in specific, the population of human and weather forecasting, plus to solve problems generated in mathematics. In mathematics, stability theory addresses the stability of solutions of trajectories of dynamical systems under small perturbations of initial conditions. Besides that, the topological dynamics is the main method used in this paper, since we studied the non-autonomous discrete dynamical systems on a topological space. Next, we present a conception of weak stability (stableness) of non-autonomous discrete dynamical systems (NADDS). We evaluate a collection of weak stable points, and survey the connection among weak stableness and shadowing property. We also consider the connection among weak stableness of a non-autonomous discrete dynamical system and its induced system. In conclusion, the results of this paper will provide some helps in the modelling of a general dynamical system.

Geometric slow–fast analysis of a hybrid pituitary cell model with stochastic ion channel dynamics

To obtain explicit understanding of the behavior of dynamical systems, geometrical methods and slow–fast analysis have proved to be highly useful. Such methods are standard for smooth dynamical systems and increasingly used for continuous, non-smooth dynamical systems. However, they are much less used for random dynamical systems, in particular for hybrid models with discrete, random dynamics. Here we propose a geometrical method that works directly with the hybrid system. We illustrate our approach through an application to a hybrid pituitary cell model in which the stochastic dynamics of very few active large-conductance potassium (BK) channels is coupled to a deterministic model of the other ion channels and calcium dynamics. To employ our geometric approach, we exploit the slow–fast structure of the model. The random fast subsystem is analyzed by considering discrete phase planes, corresponding to the discrete number of open BK channels, and stochastic events correspond to jumps between these planes. The evolution within each plane can be understood from nullclines and limit cycles, and the overall dynamics, e.g., whether the model produces a spike or a burst, is determined by the location at which the system jumps from one plane to another. Our approach is generally applicable to other scenarios to study discrete random dynamical systems defined by hybrid stochastic–deterministic models.

Rational dimension of a basis of a regression model for adaptive short-term forecasting the state of a discrete nonstationary dynamic system

Relevance. Today, there are many methodologies for predicting power consumption of various objects. However, there is no a general methodology that is suitable for all types of energy systems, including the sectoral characteristics of small northern settlements and other objects with the stochastic nature of electricity consumption schedules. At the same time, during the development of problem-oriented forecasting methods, it is necessary to take into account computational and statistical features of forecasted time series to the maximum and apply them adequately. The mentioned circumstance prompts the creation of criteria-indicators that allow evaluating the quality of the applied model for solving the forecasting problem, correctness of its construction and correctness of applying a priori information about the object and its physical properties.Aim. Develop and apply the criteria-indicators, which allow evaluating the quality of the forecast regression model and the influence of the dimensionality of such model base on a forecasting error. Methods. The choice of rational dimensionality of the regression model basis for the adaptive forecasting problem is based on the known and developed criteria-indicators. The main provisions of such criteria-indicators were formulated, which provide an assessment of the quality of conditioning of an equivalent square matrix, the presence of uninformative elements of the matrix, and linear dependence of the columns. Results. Based on the analysis of criteria-indicators, the authors selected a rational dimension of the regression model basis for the problem of adaptive short-term forecasting of the state of discrete non-stationary dynamic systems. Conclusions. The authors have previously selected the most promising criteria-indicators and developed a normalized difference factor of diagonal predominance. This allows us to evaluate the influence of the basis size change on the regression model quality when building an approach of adaptive short-term forecasting of electricity consumption by autonomous power systems of small northern settlements on the basis of regression analysis methods. Based on the analysis of criteria-indicators the authors obtained information about the influence of the regression model basis dimension on the forecasting problem solution error. The authors stated the further stages of research to reduce this error. The paper introduces and describes one of the ways to improve the forecasting model quality. The dependence of the forecasting error on the size of the regression model basis were revealed; the criteria-indicators considered in the article were successfully applied. It is confirmed that the pre-selected and developed criteria-indicators make it possible, at the stage of compiling an equivalent square matrix and performing preliminary actions on it, to track changes within the matrix. The changes will lead to improvement in the solution of the problem of adaptive short-term forecasting.

Fixed points and attractors of reactantless and inhibitorless reaction systems

Reaction systems are discrete dynamical systems that model biochemical processes in living cells using finite sets of reactants, inhibitors, and products. We investigate the computational complexity of a comprehensive set of problems related to the existence of fixed points and attractors in two constrained classes of reaction systems, in which either reactants or inhibitors are disallowed. These problems have biological relevance and have been extensively studied in the unconstrained case; however, they remain unexplored in the context of reactantless or inhibitorless systems. Interestingly, we demonstrate that although the absence of reactants or inhibitors simplifies the system's dynamics, it does not always lead to a reduction in the complexity of the considered problems.

Index Matrix-Based Modeling and Simulation of Buck Converter

The approach described in this paper handles the parameters and characteristics (analog and discrete ones) of a Buck DC-DC converter (in its power and control parts) in a common manner. The usage of probably complicated differential equations for discrete dynamical systems is avoided by means of index matrix equations, which can be easily understood. Compared to classical matrix models, the proposed index matrix models are more descriptive and also smaller in size. Such functionality is widely applied by the authors, and a new operation is defined and used as well. The relevance of the proposed techniques in power electronics, because of switching topologies and a limited numbers of components, is argued. Respective examples of functioning modes of the considered converter, in which power circuits and controllers are modeled jointly, are given. Estimations of analog values are based on partly linear dependencies, which are shown to be adequate in first-order analyses. Specific expressions for a Buck DC-DC converter are presented. A model-solving technique and an exhaustive search on a parameter space are considered in detail and automated via well-formalized algorithms. Nested parameter intervals and verifications with normal probability distributions are used in an optimization procedure. The full agreement of implementation via MATLAB source code with results obtained via Simulink is demonstrated. The short simulation times of this software (compared to Simulink and a .NET desktop application developed by the authors) justify the search for optimal variants in a wide multi-dimensional space. A max–min procedure with 10,000 simulations in each verification step is presented.

A three-dimensional model with two-body interactions for endothelial cells in angiogenesis

We introduce a three-dimensional mathematical model for the dynamics of vascular endothelial cells during sprouting angiogenesis. Angiogenesis is the biological process by which new blood vessels form from existing ones. It has been the subject of numerous theoretical models. These models have successfully replicated various aspects of angiogenesis. Recent studies using particle-based models have highlighted the significant influence of cell shape on network formation, with elongated cells contributing to the formation of branching structures. While most mathematical models are two-dimensional, we aim to investigate whether ellipsoids also form branch-like structures and how their shape affects the pattern. In our model, the shape of a vascular endothelial cell is represented as a spheroid, and a discrete dynamical system is constructed based on the simple assumption of two-body interactions. Numerical simulations demonstrate that our model reproduces the patterns of elongation and branching observed in the early stages of angiogenesis. We show that the pattern formation of the cell population is strongly dependent on the cell shape. Finally, we demonstrate that our current mathematical model reproduces the cell behaviours, specifically cell-mixing, observed in sprouts.

Experimenting with discrete dynamical systems

We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients), and more generally rational transformations of the positive orthant of R k into itself.

Symbolic Dynamics and Topological Complexity in Discrete Dynamical Systems

This paper presents a comprehensive review of symbolic dynamics and topological complexity in the context of discrete dynamical systems. We explore the fundamental concepts, recent theoretical advancements, and emerging applications of these powerful analytical tools. Symbolic dynamics, which represents system trajectories as sequences of symbols, provides a bridge between continuous dynamics and discrete mathematics. We discuss its origins, development, and key theorems, emphasizing its role in studying long-term behavior and chaos. Topological complexity, closely related to topological entropy, offers a measure of the intricacy of orbits within a dynamical system. We examine various methods for computing and analyzing topological complexity, highlighting its significance in understanding chaotic behavior and mixing properties. The interplay between symbolic dynamics and topological complexity is thoroughly investigated, demonstrating how symbolic representations facilitate the computation of topological entropy and the study of structural stability. We present recent applications of these concepts in diverse fields, including information theory, cryptography, biological systems, and ergodic theory. Case studies illustrate the practical utility of symbolic dynamics and topological complexity in solving real-world problems. Additionally, we discuss open problems and future research directions, emphasizing the potential for further theoretical developments and novel applications. This review aims to provide both an accessible introduction for newcomers and a valuable resource for established researchers in the field of discrete dynamical systems. By synthesizing current knowledge and highlighting key challenges, we hope to stimulate further research and cross-disciplinary collaborations in this rich and evolving area of mathematics.

Power maximization for a discrete-model of multistage dynamic irreversible isothermal-chemical-engine with linear mass-transfer law

Power maximization for a discrete-model of multistage dynamic irreversible isothermal-chemical-engine (IICE) with a linear mass-transfer-law is investigated by applying discrete-maximum-principle and dynamic-programming method with conditions of fixed initial-time and fixed initial key-component-concentration (KCC). Relationships among the maximum power (MPO), final-KCC and process-period for discrete multistage dynamic IICE system are discussed in detail. Results show that when final-concentration and process-period are fixed, there does exist an optimal control-strategy for MPO, and internal-irreversibility factor has a significant effect on the MPO, but the corresponding optimal-concentration-configurations are the same. When final-time and final-concentration are fixed and free, respectively, there does exist an optimal final-KCC for MPO, and change of internal-irreversibility factor has effects on MPO and the corresponding optimal-KCC configurations now; the more total stage-number of chemical-engines is, the more closely optimization results of discrete-model tend to those of the corresponding continuous-model. When final-time and final-concentration are free and fixed, respectively, numerical example shows that for the discrete multistage dynamic IICE system with total stage-numbers of N=25 and N=50, respectively, total MPOs are with relative errors of 1.66% and 3.33%, respectively, comparing with that for the continuous-model of multistage dynamic IICE system. Both the model established and the mothed adopted herein are effective.

A criterion of periodic point stability based on Lyapunov exponent

The logistics equation is the most classical model of population growth. Influenced by external environmental factors and growth inertia, the total population is in a state of periodic equilibrium. so, studying the stability of the periodic solution of the logistics equation is an important issue. If the logistics equation is considered as a function, the general method to judge the stability of the periodic point is to bring in the derivative of the function after iterating n times to take the value. The Lyapunov exponent is originally an important method used to judge the stability of dynamical systems. If the logistics map is considered as a discrete dynamical system, applying the Lyapunov exponent to the determination of the periodic solution will largely reduce the computational effort.

Category of Discrete Dynamical System

A dynamical system is a method that can describe the process, behavior, and complexity of a system. In general, a dynamical system consists of a discrete dynamical system and a continuous dynamical system. This dynamical system is very interesting if seen from the algebraic side. One of them is about category theory. Category theory is a very universal theory in mathematical concepts. In this research, the dynamical system used is a discrete dynamical system represented as a directed graph with nodes in the graph called states. This discrete dynamical system has a height which is shown on the dynamical map in which the number of states at each height is called a profile. In this research, it will be proved whether the discrete dynamical system with the same profile is a category. Also, why category theory is needed in discrete dynamical systems will be investigated. The result of this study shows that the discrete dynamical system with the same profile is a category with its morphism is an evolution from one state to another state in different dynamical systems. Furthermore, category theory is needed for discrete dynamical systems to know about the properties and structure of discrete dynamical system.

A predictive TDM-PON resource allocation using the ANN method based on equilibrium points of discrete dynamical systems

Resource allocation is one of the challenges in the time division multiple access (TDM) passive optical network (PON) area. Much research has been done to find an efficient resource allocation method. Using machine learning (ML) and finding proper ML algorithms in predictive dynamic bandwidth allocation remains an open research question. This paper aims to predict resource allocation accurately and efficiently based on historical requests. Artificial neural networks (ANN) with historically allocated bandwidth from different time series are used to build the prediction model. This model is constructed by the normal equation method. The proposed model needs a small dataset for its training phase, while its accuracy is higher than other models. The discrete dynamical system builds to find an equilibrium point, which is a stable point in the system. The proposed model can be implemented as a pre-training, or it can be dynamically refined when the model is running by using some of its output as feedback. The system evaluation verifies that the proposed DBA (EQ-ANN) elevates the quality of service (QoS) metrics, such as system throughput, packet drop probability, and packet means delay. The results show that the system throughput improves by up to 5%, and packet loss improves by up to 15%. Moreover, the prediction accuracy of the system reached 99.48%.

Anomaly detection and prediction evaluation for discrete nonlinear dynamical systems

Anomalies in dynamical systems mostly occur as deviations between measurement and prediction. Current anomaly detection methods in multivariate time series often require prior clustering, training data, or cannot distinguish local and global anomalies. Furthermore, no generalized metric exists to evaluate and compare different prediction functions regarding their amount of anomalous behavior. We propose a novel methodology to detect local and global anomalies in time series data of dynamical systems. For this purpose, a theoretical density distribution is derived assuming that only noise conceals the time series. If the theoretical and the empirical density distribution yield significantly different entropies, an anomaly is assumed. For a local anomaly detection, the Mahalanobis distance using the theoretical noise distribution’s covariance is applied to evaluate sequences of predictions and measurements. In addition, the Wasserstein metric enables a comparison of predictions using the distance between the noise and empirical distribution as a measure for selecting the best prediction function. The proposed method performs well on nonlinear time series such as logistic growth and enables a useful selection of a prediction model for satellite orbits. Thus, the proposed method improves anomaly detection in time series and model selection for nonlinear systems.

Application of Migrating Optimization Algorithms in Problems of Optimal Control of Discrete-Time Stochastic Dynamical Systems

The problem of finding the optimal open-loop control for discrete-time stochastic dynamical systems is considered. It is assumed that the initial conditions and external influences are random. The average value of the Bolza functional defined on individual trajectories is minimized. It is proposed to solve the problem by means of classical and modified migrating optimization algorithms. The modification of the migrating algorithm consists of cloning the members of the initial population and choosing different strategies of migratory behavior for the main population and for populations formed by clones. At the final stage of the search for an extremum, an intensively clarifying migration cycle is implemented with the participation of three leaders of the populations participating in the search process. Problems of optimal control of bundles of trajectories of deterministic discrete dynamical systems, as well as individual trajectories, are considered as special cases. Seven model examples illustrating the performance of the proposed approach are solved.

Caputo and Conformable Fractional Order Guava Model for Biological Pest Control: Discretization, Stability and Bifurcation

Abstract Two predator-prey model describing the guava borers and natural enemies are studied in this paper. Positivity, existence, and uniqueness of the solution, global and local stability analysis of the fixed points of the first model based on the Caputo fractional operator are studied. By adding piecewise constant functions to the second model including conformable fractional operator allows us to transition discrete dynamical system via discretization process. Applying Schur-Cohn criterion to the discrete system, we hold some regions where the equilibrium points in the discretized model are local asymptotically stable. We prove that discretized model displays supercritical Neimark–Sacker bifurcation at the equilibrium point. Theoretical and numerical results show that the discretized system demonstrates richer dynamic properties such as quasi-periodic solutions, bifurcation, and chaotic dynamics than the fractional order model with Caputo operator. All theoretical results are interpreted biologically and the optimum time interval for the harvesting of the guava fruit is given.