This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous...


 Em matemática, o termo "não linear" geralmente corresponde a uma análise mais difícil. Uma vez que os sistemas lineares são simples de analisar, maneira importante entender não é descobrir em condições eles podem ser bem aproximados por lineares. A respeito disso, temos um célebre teorema equações diferenciais Grobman [Gro59] e Hartman [Har60a], nos garante campo vetorial classe C1, X : ...

Variability in the underlying time structure of dynamical systems and its consequences are encountered in a variety of mathematical contexts, though are not always recognized as such. An obvious example is the numerical time-discretization of ordinary differential equations, where the relationship between the behaviour of an approximate discrete-time dynamical system and that of the original co...

For a non-autonomous dynamics with discrete time defined by tempered sequence of upper-triangular matrices, we obtain lower and upper bounds for the Grobman regularity coefficients. We also give two applications these results: an bound coefficients exterior powers sequence, simple proof Oseledets' multiplicative ergodic theorem cocycles over measure-preserving transformation without using Kingm...

Contents Chapter 1. Introduction 7 1. Historical sketch 7 a. Homoclinic tangles 7 b. Geodesic flows 8 c. Picking up from Poincaré 8 d. Modern hyperbolic dynamics 9 e. The slowness of the initial development 9 2. Hyperbolic dynamics 10 3. Outline of this survey 10 a. Regularity 11 Chapter 2. Hyperbolic sets and stable manifolds 13 1. Definitions and examples 13 a. Hyperbolic linear maps, adapted...

Discretizationsand Grobman-HartmanLemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For one-step methods , it is proved that the linearizing conjugacy for ordinary diierential equations in Grobman-Hartman Lemma is, with decreasing stepsize, the limit of the linearizing conjugacies of the discrete systems obtained via time-discretizations. Similar r...