Dynamical Systems with Applications using Python

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

An introduction to dynamical systems

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.

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].

1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation, 1997

C o 1x1 p u t at i o ii by D y ii alii i c a1 S y s t e 111 s

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.

Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, 2017

Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

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.

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.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, 2017

Control of chaos teaches that control theory can tame the complex, random-like behaviour of chaotic systems. This alliance between control methods and physics-cybernetical physics-opens the door to many applications, including dynamics-based computing. In this article, we introduce nonlinear dynamics and its rich, sometimes chaotic behaviour as an engine of computation. We review our work that has demonstrated how to compute using nonlinear dynamics. Furthermore, we investigate the interrelationship between invariant measures of a dynamical system and its computing power to strengthen the bridge between physics and computation.This article is part of the themed issue 'Horizons of cybernetical physics'.