Minimal attractor embedding dimension for discrete dynamic system using state-space method: theoretical ground

A new nonlinear method is proposed for predicting the mine water inrush. According to the fractal theory, the phase space of time series obtained from the mine water inrush is reconstructed. And the minimum embedding dimension is determined with Cao method, then the minimum embedding dimension is used for the input node of the support vector machines. The prediction model of time series is established based on the support vector machines. The mine water inrush of new well in Fang Shan mine is predicted with this method. And the prediction results show that they coincide with the observing values better. Different dimensions is chosen to predict the mine water inrush, and when the minimum embedding dimension is four, its prediction precision is the highest. Comparing with the predicting results of three methods including least square method, index fitting method and SVM method, the prediction precision of SVM is the best. It proves that the method put forward in this paper is reliable to predict the mine water inrush.

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.

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.

Discrete dynamical systems - generalities on dynamical systems, examples of discrete dynamical systems, problems one-dimensional dynamical systems - preliminaries on one-dimensional systems, nonlinear one-dimensional systems, global behaviour and conjugacy, parameter space analysis, problems functions and matrices - matrices, matrices as linear operators, derivative and first order approximation, problems discrete linear dynamical systems - linear and affine processes, stability and instability of the origin, the origin as a saddle point, Eigenvalues with modulus I, problems nonlinear dynamical systems - global stability of equilibrium points, sinks, repellers and saddles, bounded invariant sets, bifurcation, problems chaotic behaviour - attractors, chaotic dynamical systems, dimension, Lyapunov exponents, problems analysis of four dynamical systems - a blood cell population model, predator-prey models, the Lorenz model, neural networks, problems.

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.

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.

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.

Realizing Euclidean distance matrices by sphere intersection

The analysis of observed time series from nonlinear systems is usually done by making a time-delay reconstruction to unfold the dynamics on a multi-dimensional state space. An important aspect of the analysis is the choice of the correct embedding dimension. The conventional procedure used for this is either the method of false nearest neighbors or the saturation of some invariant measure, such as, correlation dimension. Here we examine this issue from a complex network perspective and propose a recurrence network based measure to determine the acceptable minimum embedding dimension to be used for such analysis. The measure proposed here is based on the well known Kullback-Leibler divergence commonly used in information theory. We show that the measure is simple and direct to compute and give accurate result for short time series. To show the significance of the measure in the analysis of practical data, we present the analysis of two EEG signals as examples.

An experimental time series from a tensile test of an Al-Mg alloy in the serrated plastic flow domain is analyzed for signature of chaos. We employ state space reconstruction by embedding of time delay vectors. The minimum embedding dimension is found to be 4 and the largest Lyapunov exponent is positive, thereby providing prima facie evidence for chaos in an experimental time series of serrated plastic flow data. \textcopyright{} 1996 The American Physical Society.

Fault diagnosis of key components of high-speed train is an important issue to guarantee safety operation of high-speed train. In this paper, chaotic method based on reconstructed phase space is introduced for analyzing fault information of high-speed train accounting for nonlinearity, aperiodicity and nonrandomness of vibration signal of high-speed train. Firstly, the optimal delay time and the minimum embedding dimension are determined by mutual information method and Cao's method respectively in order to reconstruct phase space of vibration signals of high-speed train. And then the pseudo-phase trajectories and correlation dimension are analyzed to determine the fault information and failure states of critical components. The analysis shows the vibration signals of high-speed train have chaotic motion characteristics, and their attractors in reconstructed phase space have the similar movement structure. In addition, the areas of pseudo-phase trajectories change gradually with the damping changing of the secondary lateral damper on high-speed train. The results demonstrate that chaotic method is an effective method for analyzing fault information of high-speed train.

The set of trajectories for discrete dynamical systems (DDS) in the space is investigated. These sets are the solutions for difference equations in a metric space (space of nonempty convex compacts with the Hausdorff metric). On the basis of the comparison principle the general theorems on stability in terms of two measures were established. Applying the Minkowskij theory of mixed volumes, for some classes of nonlinear DDS in space the finite-dimensional comparison systems were constructed. The stability in terms of two measures and Lyapunov stability of fixed points for DDS in space were studied. The examples of studies of certain dynamical systems were given to illustrate the effectiveness of obtained results.

Measures of LLE of heart rate in different frequency bands: a possible measure of relative vagal and sympathetic activity

Estimating the number of reaction intermediates for formaldehyde oxidation by analysis of the chaotic behavior

Local exponential divergence plot and optimal embedding of a chaotic time series