An iterative algorithm for performance evaluation of Stochastic Event Graph

In this paper we introduce the notion of regular ordering for periodic sequences based on the gaps between the entries. We define the notion of regular preserving functions using Schur convexity. This is used to extend some optimization results in queuing control problems. In particular, we show that the maximal traveling time in a stochastic event graph as well as the transmission times in a channel with redundancy, decrease (in the stochastic sense) when the input sequence becomes more regular.

The traditional models of distributed computing systems employ the mathematical formalisms of discrete event dynamical systems along with Petri Nets. However, such models are complex to analyze and computationally expensive. Interestingly, the evolving distributed computing systems closely resemble the discrete dynamical systems. This paper constructs and analyzes the model of distributed computing systems by applying mathematical formalisms of the discrete dynamical systems. The proposed model embeds the partial ordering of states under happened-before relation in the set of globally observable states. The stability of the proposed model is analyzed using the first order linear approximation.

With the rapid development of computer science, there are more and more kinds of discrete dynamic systems. Computer integrated system CIMS, network communication database administrator system, and human behavior analysis system are all discrete dynamic systems. At present, many researchers have studied by adding human behavior data to discrete dynamic systems. This paper aims to study the behavior data of English learners by using the discrete dynamic modeling technology of complex systems and the discrete dynamic system modeling method of Petri nets. By adding the behavior data of learners to the discrete dynamic system of fuzzy Petri nets, the system is diagnosed and optimized. The experimental results show that the complex discrete dynamic system in this paper has achieved good experimental results according to the performance indicators selected in theory. Based on the combination of the above technologies and systems, the fuzzy Petri net discrete dynamic system studied in this paper improves the processing speed of English learners' behavior data.

Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a collection $T_1, T_2, \ldots, T_m \subseteq \mathbb{R}^d$ of semialgebraic sets, we can associate with each $T_i$ an atomic proposition $P_i$ which evaluates to true at time $n$ if, and only if, $M^ns \in T_i$. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system $(M,s)$ and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

As a way of resolving vehicle congestion, there is a feedback control approach which models a traffic network as a discrete dynamical system and derives feedback gain for controlling green light times of each junction. Since the input is the sensory observed traffic flow of each link, and since the state equation models both the topology and the parameters of the network, it is effective for adaptive control of a wide area traffic in real-time. One of the essential factors in a state equation is the vehicles’ turning ratio at each junction. However, in a normal traffic sensor layout, it is impossible to directly measure this value in real-time, and values from traffic census are used. This paper is to propose a method that predicts this value in real-time through machine learning and gives more appropriate feedback control. Out idea is to find the turning ratio through probabilistic search by Reinforcement Learning referring to the degree of improvement of the entire traffic flow. At this moment we have finished formulation of the scheme and the verification for the performance by a traffic simulator is on the way.KeywordsReinforcement LearningTraffic light ControlFeedback ControlDiscrete Dynamical SystemSplit

We discuss the possibility to formulate the dynamics of spin states described by the Schr¨odinger equation for pure states and the von Neumann equation (as well as the GKSL equation) for mixed states in the form of quantum kinetic equations for probability distributions. We review an approach to the spin-state description by means of the probability distributions of dichotomic random variables.

Robust control of discrete time uncertain dynamical systems

systems. In this paper, we firstly study the Noether symmetry and Noether exact invariants of discrete nonconservative dynamical systems. Secondly, we study perturbation to symmetry and adiabatic invariants of discrete dynamical systems. Finally, we give an example to illustrate the application of these results.

This paper focuses on discrete dynamic optimization on train rescheduling on single-track railway with the consideration of train punctuality and station satisfaction degree. A discrete dynamic system is firstly described to mimic train rescheduling, and a state transition function is specially designed according to the train departure event. The purpose of this function is to improve simulation efficiency by directly confirming the next discrete time. After the construction and analysis of optimization models to discrete dynamic system, a two-stage heuristic search strategy is developed, by using clustering hierarchy theory and stochastic search strategy, to obtain train departure time and arrival time before each state transition of the system. Finally, a numerical experiment is conducted to verify the proposed system, models, and the heuristic search strategy. The result shows that the discrete dynamic system, together with the state transition function and heuristic search strategy, shows better performance in simulation efficiency and solution quality.

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

In this article, performance of sliding frame (SF) R-ALOHA protocols for real-time distributed wireless networks is investigated analytically and by simulation. First, a discrete Markov chain and Monte Carlo modeling are constructed to evaluate dynamic behavior of the protocol in transient state including distribution of the system stabilization time (SST) and the average number of successful terminals. Furthermore, a hierarchical decomposition is conducted to simplify steady state analysis, thus generating two one-dimensional Markov chains for closed-form performance of SF R-ALOHA under local wireless environment. Terminals with Poisson message arrivals and Poisson message length are analyzed. Consequently, performance indices, such as throughput, the average message delay, and packet dropping probability, are derived from the proposed analytic models. Capture effects on the SF R-ALOHA system in the presence of multipath and shadowing are also obtained by the Markov chain analysis. The numerical results from the analytic models are compared with that from simulation and equilibrium point analysis, proving correctness, accuracy, and scalability of the decomposition. The results also reveal performance characteristics of the SF R-ALOHA system.

Multi-transitivity in non-autonomous discrete systems

Effects of squeezed vacuum on transient optical response in a ∧ system