Notion Of Dynamical System Research Articles

A new approach to the entropy of a transitive BE-algebra with countable partitions

The concept of entropy and information gain of BE-algebras in scientific disciplines such as information theory, data science, supply chain and machine learning assists us to calculate the uncertanity of the scientific processes of phenomena. In this respect the notion of filter entropy for a transitive BE-algebra is introduced and its properties are investigated. The notion of a dynamical system on a transitive BE-algebra is introduced. The concept of the entropy for a transitive BE-algebra dynamical system is developed and, its characteristics are considered. The notion of equivalent transitive BE-algebra dynamical systems is defined, and it is proved the fact that two equivalent BE-algebra dynamical systems have the same entropy. Theorems to help calculate the entropy are given. Specifically, a new version of Kolmogorov– Sinai Theorem has been proved. The study introduces the concept of information gain of a transitive BE-algebra with respect to its filters and investigates its properties. This study proposes the use of filter entropy to approximate the level of risk introduced by a BE-algebra dynamical system. This aim is reached by defining the information gain with respect to the filters of a BE-algebra. This methodology is well developed for use in engineering, especially in industrial networks. This paper proposes a novel approach to assess the quantity of uncertainty, and the impact of information gain of a BE-algebra dynamical system.

Properness of foliations

In this paper, we consider the question whether the leaf space of a foliation on a manifold is T0 if and only if each leaf is proper. In general, the answer is negative because there is a codimension one foliation of a non-paracompact 3-dimensional manifold constructed by Milnor which consists of only one leaf, which is non-proper. However, using the notions of dynamical systems, we show that the answer for paracompact manifolds is affirmative. In other words, a foliation F on a paracompact manifold M is proper if and only if the leaf space M/F is T0. In fact, a leaf L of a foliation on a paracompact manifold is proper if and only if its leaf class (i.e. the subset of points whose leaf closures coincide with the closure of L) consists of only L itself. To show this, we show that the subset of leaves contained in the leaf class of a non-proper leaf L∈F of a foliation F on a paracompact manifold for has the cardinality of the continuum. In addition, similar results hold for flows and homeomorphisms.

Compressibility, Laws of Nature, Initial Conditions and Complexity

We critically analyse the point of view for which laws of nature are just a mean to compress data. Discussing some basic notions of dynamical systems and information theory, we show that the idea that the analysis of large amount of data by means of an algorithm of compression is equivalent to the knowledge one can have from scientific laws, is rather naive. In particular we discuss the subtle conceptual topic of the initial conditions of phenomena which are generally incompressible. Starting from this point, we argue that laws of nature represent more than a pure compression of data, and that the availability of large amount of data, in general, is not particularly useful to understand the behaviour of complex phenomena.

Mathematical Approach to Strokes as an Attractor Within Communication Dynamical System

Transactional Analysis as a personality theory has offered some powerful concepts to explain and improve communication between individuals. On the other hand, positive productive communication among large numbers of people, as a compound set of transactions, as an essential aspect of human survival has not been so well explained. Like the weather and other chaotic processes, group behavior is not easily understood or predicted. It has been long suspected that in large groups of people (organizations, communities, societies), positive communication has a leading role in maintaining the duration and quality of communication. It is the aim of this paper to relate the mathematics notion of dynamical systems to the compound system of communication. The postulate that strokes, a concept introduced by E. Berne as a way in which people recognize each other, and elaborated by Steiner as a way of exchanging information, is discussed as a concept that introduces stability into the functioning of large groups.

Phase-locked trajectories for dynamical systems on graphs

We prove a general result on the existence of periodic trajectories of systemsof difference equations with finite state space which are phase-locked oncertain components which correspond to cycles in the coupling structure. A maintool is the new notion of order-induced graph which is similar in spirit to aLyapunov function. To develop a coherent theory we introduce the notion ofdynamical systems on finite graphs and show that various existing neuralnetworks, threshold networks, reaction-diffusion automata and Boolean monomialdynamical systems can be unified in one parametrized class of dynamical systemson graphs which we call threshold networks with refraction. For an explicitthreshold network with refraction and for explicit cyclic automata networkswe apply our main result to show the existenceof phase-locked solutions on cycles.

The foundational crisis of cognitive science: challenging the emergentist challenge

The following pages contend that, in spite of its intensive development, contemporary cognitive science has recently entered a phase of fairly acute uncertainty and confusion regarding some of its most essential foundations. They emphasize two aspects of this foundational crisis, specifically vindicating the existence of a crisis of naturalism and of a crisis of representationalism. Like any foundational crisis, this situation constitutes a serious threat to the significance of the empirical achievements of cognitive science. A threat calling for renewed efforts to provide it with secure foundations that can only be obtained through acloser collaboration between empirical and foundational investigations, or, more concretely, between cognitive scientists and philosophers. They also outline a general strategy to address this threat, and illustrate itabout one aspect of particular importance of the naturalist crisis, namely the emergentist challenge to the orthodoxy of non reductive functionalism. They argue for the rejection of one version of this emergentist challenge,and they lay out a minimal condition that any other version of emergentism must meet. It is still unclear whether this condition is yet satisfactorily met by some version, and in particular by the sort of emergentism associated with the notion of dynamical system. Clarifyingthis issue should accordingly be seen as a top priority on the agenda of cognitive philosophy.

Linear algebra algorithms as dynamical systems

Any logical procedure that is used to reason or to infer either deductively or inductively, so as to draw conclusions or make decisions, can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state develops into another state by following a certain specific rule. Such an action is generally formalized as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyse, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in submanifolds of matrices are cast in fairly general frameworks, of which special cases have been found to afford unified and fundamental insights into the structure and behaviour of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

Generalizing Mutational Equations for Uniqueness of Some Nonlocal 1st-Order Geometric Evolutions

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ N evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

Dissipativity theory and stability of feedback interconnectionsfor hybrid dynamical systems

In this paper we develop a unified dynamical systems framework for a general class of systems possessing left‐continuous flows; that is, left‐continuous dynamical systems. These systems are shown to generalize virtually all existing notions of dynamical systems and include hybrid, impulsive, and switching dynamical systems as special cases. Furthermore, we generalize dissipativity, passivity, and nonexpansivity theory to left‐continuous dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended to left‐continuous dynamical systems providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous‐time dynamics, and dissipated energy over the resetting events. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback interconnections of left‐continuous dynamical systems. These results generalize the positivity and small gain theorems to the case of left‐continuous, hybrid, and impulsive dynamical systems.

Methods for the investigation of dynamical systems with impulse action and ?mortal? dynamical systems

The notions of dynamical systems with impulse action and “mortal” dynamical systems are introduced. Their connection with the idealizations of ordinary dynamical systems is considered. General methods for the investigation of these systems are worked out.

Book review:Chaotic behavior of deterministic dissipative systems

This book contains a survey of theoretical and experimental aspects of the chaotic behavior of deterministic systems. It is an introductory text aimed at the level of graduate students and researchers from a variety of fields (mathematics, physics, biology, and engineering). The monograph provides a comprehensive bibliography on the subject up to the date of publication. In my opinion, one of the most interesting features of the book is its detailed description of experiments in which chaos constitutes a significant backdrop. The book consists of seven chapters and two technical appendices. The introductory Chapter 1 discusses the significance of chaos as a model of many seemingly random processes in nature and defines the class of dissipative systems described in greater detail later in the book. Chapter 2 discusses basic notions of dynamical systems and their asymptotic behavior, including the definitions of chaos and strange attractors. Chapter 3 deals with the transition from order to chaos, illustrated by a number of examples. Chapter 4 discusses numerical methods for studies of parametric dependences, bifurcations, and chaos. Chapters 5 and 6 survey experimental observations of chaotic behavior, including a number of experiments and data from mechanical and electromechanical systems, semiconductors, lasers, chemical and biological systems, and hydrodynamics. Some of the experiments and data are based on the authors' work. Finally, Chapter 7 deals with spatiotemporal behavior in distributed systems, presenting a brief review on results in the areas of cellular automata, coupled map lattices, and partial differential equations:

Invariance and stability of fuzzy systems

The elaboration of a stability theory of fuzzy systems is fully justified by the use, in a variety of practical situations, of fuzzy controllers [ 11, 27, 28). However, such a theory may be founded upon ideas that arise from a historical study of dynamical systems theory. During the last quarter of the nineteenth century, the study of ordinary differential equations underwent some rather radical changes. Prior to this period, the major emphasis in the subject had been on the methods of solving equations. It was during this period that Peano [20] gave a rigorous proof of the existence theorem for solutions of ordinary differential equations. During the same period, Lipschitz [ 151 and Picard [22] showed that the method of successive approximations led to a proof of the existence and the uniqueness of solutions to the initial value problem. These developments represent the dhuement of the attempts to solve differential equations. As the works of Peano, Lipschitz and Picard were finishing one chapter in the book of differential equations, another was being initiated with the research of Liapunov [ 141 and Poincare [23]. This new chapter was based on an entirely different approach, where one did not attempt to compute the solutions; rather one tried to exploit their topological properties. Poincart, in his famous m&moire [23], was the first to look at an ordinary differential equation from the point of view of the geometry of the trajectories, thereby creating the topological theory of ordinary differential equations. Almost at the same time, the stability theory created by Liapunov [ 141 although heavily quantitative in its methods, stressed the importance of some topological features of ordinary differential equations. During the 50 years following Liapunov and Poincare, many important advances were made. It is with the works of Birkhoff [4], Nemytskii and Stepanov [ 191, and Smale [26] that researchers realized that the essence of this theory was based on the notion of dynamical system. Based on Liapunov’s original works, “classical” stability theory is concerned with the equilibrium points of the system and the dynamical

A FORMALIZATION OF SOME SIMULATION LANGUAGE CONCEPTS

A formalization of some simulation modeling concepts is described, which is used as a conceptual framework among simulationists and in the educational process. The notion of dynamic system is hierarchically constructed from attributes and classes: the domains of the classes are formed by elements with “similar” attributes and a set of classes is a system if it satisfies certain axioms. A simulation model is composed of two systems and four mappings also satisfying axioms. This conception reflects the capability of simulation languages to not only describe the dynamics of systems but also their structure. The theory is a framework for any sort of simulation models (discrete, continuous, combined, hybrid, analogue, galvanical, aerodynamical etc.) and allows a classification of the set of all simulation languages according to their semantics.

Lagrangian and Hamiltonian formalisms: An analysis of classical mechanics on tangent and cotangent bundles

The notion of dynamical system in classical mechanics is built starting from the trajectories on the configuration space. Classes of equivalent vector fields on tangent and cotangent bundles are obtained. This leads to a re-examination in detail of the meaning of the Hamiltonian and Lagrangian functions.