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2012, International Journal of Applied Mathematical Research
Dynamical system is a young and vigorously growing area of research which promises enormous potential and opportunities. This paper aims to introduce some of the preliminary concepts of dynamical systems. Many application sides of the subject are noted to encourage the reader for future developments. Resources are supplied in the references for further reading.
2010
The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems
2019
This is an english version of the notes written for my lectures on "Tópicos de Sistemas Dinâmicos" for the "Licenciatura em Matemática" of the University of Minho, during the last decade (available at my page ). Emphasis is on examples, and on the interplay between different areas of mathematics. Some very important parts of the modern theory of dynamical systems, as hyperbolic theory, hamiltonian systems, or the qualitative theory of differential equations, are almost completely missing. Other interesting results or directions are only sketched. Main references and sources are [KH95, HK03], others are suggested along the text. e.g. means exempli gratia, that is, "for example", and is used to introduce important or interesting examples. ex: means "exercise", to be solved at home or in the classroom. indicates the end of a proof. Pictures were made with Grapher on my MacBook, or taken from Wikipedia, or produced with my own Java codes, like the one below.
2007
One approach starts from time-continuous differential equations and leads to time-discrete maps, which are obtained from them by a suitable discretization of time. This path is pursued, eg, in the book by Strogatz [Str94]. 1 The other approach starts from the study of time-discrete maps and then gradually builds up to time-continuous differential equations, see, eg,[Ott93, All97, Dev89, Has03, Rob95].
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set-one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.
Abstract These are easy-to-read lecture notes for a short first-year Ph. D. student course on Applied Dynamical Systems, given at the London Taught Course Centre in Spring 2008, 2009 and 2010. The full course consists of two parts, covering four and six hours of lectures, respectively. The first part taught by Wolfram Just is on Basic Dynamical Systems. The present notes cover only the second part, which leads From Deterministic Chaos to Deterministic Diffusion.
2012
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Akademos : Revista de Ştiinţă, Inovare, Cultură şi Artă, 2022
State university of Moldova sistemele dinamice neautonome şi aPlicațiile lor rezumat. Articolul reprezintă o scurtă trecere în revistă a cercetărilor efectuate de autor în ultimii 10-15 ani privind sistemele dinamice neautonome și aplicațiile acestora. Sistemele dinamice neautonome constituie un nou domeniu ce contribuie la dezvoltarea rapidă a matematicii (teoria sistemelor dinamice). Mii de articole, inclusiv zeci de articole de sinteză și un șir de monografii despre sistemele dinamice neautonome au fost publicate în ultimele decenii, iar problematica respectivă a făcut cap de afiș la conferințele internaționale. Autorul a publicat trei monografii pe problema sistemelor dinamice neautonome. În acest articol este oferită o prezentare generală a rezultatelor obținute. Cuvinte-cheie: soluții periodice, soluții cvasi-periodice, soluții aproape periodice Bohr/Levitan, soluții Bohr aproape automorfe, soluții recurente Birkhoff, soluții stabile Lagrange, soluții aproape recurente, soluții stabile Poisson, stabilitate Lyapunov, stabilitate asimptotică, atractori globali. summary. This article is devoted to a brief overview of the author's works over the past 10-15 years on non-autonomous dynamic systems and their applications. Non-autonomous dynamical systems are a new and rapidly developing field of mathematics (theory of dynamical systems). Thousands of articles, dozens of reviews and a number of monographs on non-autonomous dynamic systems and their applications have been published over the past 10-15 years. Special international conferences and scientific journals are dedicated to them. My results on non-autonomous dynamical systems and their applications are published in three monographs. In this article, we provide an overview of these results.
An introduction to dynamical systems
2007
It is in principle possible to develop the full theory of both from either perspective, but for the bulk of this course, we shall follow the latter route. This allows a generally more simple way of introducing the important concepts, which can usually be carried over to a more complex and physically realistic context.
In The Foundations of Mechanics, Chapter Five, we presented a concise summary of modern dynamical systems theory from 1958 to 1966. In this short note, we update that account with some recent developments.
Bifurcations and Periodic Orbits of Vector Fields, 1993
We present several topics involving the computation of dynamical systems. The emphasis is on work in progress and the presentation is informal-there are many technical details which are not fully discussed. The topics are chosen to demonstrate the various interactions between numerical computation and mathematical theory in the area of dynamical systems. We present an algorithm for the computation of stable manifolds of equilibrium points, describe the computation of Hopf bifurcations for equilibria in parametrized families of vector fields, survey the results of studies of codimension two global bifurcations, discuss a numerical analysis of the Hodgkin and Huxley equations, and describe some of the effects of symmetry on local bifurcation.
2013
Abstract. This paper is meant to provide an introduction to the ideas of dynamical systems. In order to do this, we must first provide an overview of measure theory and
2018
Please cite the published version birkhauser-science.de 1st ed. 2018, XVI, 665 p. 277 illus., 118 illus. in color.
Iconic Research And Engineering Journals, 2018
In this paper, we discuss some applications of exciting fields like Mathematics, Statistics, Medical, Biological Sciences, Engineering, Economics etc. We show their real-life applications both theoretically and analytically considering some phenomena of the nature. We mainly focused on some applications of dynamical system in real life cases known as chaos, iteration, fractal geometry, Mandelbrot and Julia sets. We derive some mathematical formula concerning dynamical systems. Necessary programs are considered for all cases. We use Mathematica and MATLAB to perform programming.
Journal of the Institute of Engineering
Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization. Journal of the Institute of Engineering, 2018, 14(1): 35-51
The previous system is said to be autonomous since it is not exposed to external influences. Non-autonomous systems do have external inputs.
Nonlinear Control Systems Design 1992, 1993
Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. From this point of view the long tenn behavior of control systems, such as limit sets, Mone sets. approximations on the entire time axis, ergodicity, Lyapunov exponents, stable and unstable manifolds etc. becomes accessible. This paper presents some of the underlying theory, as well as applications 1.0 the global characterization of control systems with bounded control range that are not completely controllable, to control of chaotic systems, and to exponential stability of uncertain systems.
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