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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
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.
International Journal of Applied Mathematical Research, 2012
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.
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].
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.
Nonlinear Analysis: Theory, Methods & Applications, 1978
An introduction to dynamical systems
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.
2012 IEEE International Conference on Control Applications, 2012
Nagumo's condition, also known as the subtangency condition, characterizes the set invariance with respect to the trajectories of dynamical systems. The general formulation of Nagumo's condition considers a comprehensive framework, referring to time-variant nonlinear systems and generic sets with arbitrary shapes. For time-variant linear dynamical systems and sets defined by arbitrary norms, we prove that Nagumo's condition has an equivalent form expressing the nonpositiveness of the system-matrix measure. This equivalent form is more attractive for application than the general formulation, since it points out the role of the timedependent model coefficients. We also show that, for timeinvariant linear systems and some particular cases of sets defined by weighted p-norms, {1, 2, } p ∈ ∞ , Nagumo's condition is equivalent with well-known algebraic properties (Lyapunov inequality, location of Gershgorin's disks) satisfied by the constant system-matrix. Finally, we evaluate the key ideas of the paper as a potential resource for developing educational materials dedicated to advanced studies in Control Engineering.
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
Proceedings of the National Academy of Sciences, 1967
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.
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.
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.
2006
Linear algebra plays a key role in the theory of dynamical systems, and concepts from dynamical systems allow the study, characterization and generalization of many objects in linear algebra, such as similarity of matrices, eigenvalues, and (generalized) eigenspaces. The most basic form of this interplay can be seen as a matrix A gives rise to a continuous time dynamical system via the linear ordinary differential equation ẋ = Ax, or a discrete time dynamical system via iteration xn+1 = Axn. The properties of the solutions are intimately related to the properties of the matrix A. Matrices also define nonlinear systems on smooth manifolds, such as the sphere S in R, the Grassmann manifolds, or on classical (matrix) Lie groups. Again, the behavior of such systems is closely related to matrices and their properties. And the behavior of nonlinear systems, e.g. of differential equations ẏ = f(y) in R with a fixed point y0 ∈ R d can be described locally around y0 via the linear differenti...
Journal of Mathematical Analysis and Applications, 1975
In the first part of this paper we study dynamical systems from the point of view of algebraic topology. What features of all dynamical systems are reflected by their actions on the homology of the phase space? In the second part we study recent progress on the conjecture that most partially hyperbolic dynamical systems which preserve a smooth invariant measure are ergodic, and we survey the known examples. Then we speculate on ways these results may be extended to the statistical study of more general dynamical systems. Finally, in the third part, we study two special classes of dynamical systems, the structurally stable and the affine. In the first case we study the relation of structural stability to entropy, and in the second we study stable ergodicity in the homogeneous space context.
IFAC Proceedings Volumes, 1975
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