tailieunhanh - Ebook Ordinary differential equations and dynamical systems: Part 2
(BQ) Part 2 book "Ordinary differential equations and dynamical systems" has contents: Dynamical systems, planar dynamical systems, higher dimensional dynamical systems, local behavior near fixed points, discrete dynamical systems, discrete dynamical systems in one dimension, chaos in higher dimensional systems, periodic solutions. | Part 2 Dynamical systems Author's preliminary version made available with permission of the publisher, the American Mathematical Society Author's preliminary version made available with permission of the publisher, the American Mathematical Society Chapter 6 Dynamical systems . Dynamical systems You can think of a dynamical system as the time evolution of some physical system, such as the motion of a few planets under the influence of their respective gravitational forces. Usually you want to know the fate of the system for long times, for instance, will the planets eventually collide or will the system persist for all times? For some systems (., just two planets) these questions are relatively simple to answer since it turns out that the motion of the system is regular and converges, for example, to an equilibrium. However, many interesting systems are not that regular! In fact, it turns out that for many systems even very close initial conditions might get spread far apart in short times. For example, you probably have heard about the motion of a butterfly which can produce a perturbance of the atmosphere resulting in a thunderstorm a few weeks later. We begin with the definition: A dynamical system is a semigroup G with identity element e acting on a set M . That is, there is a map T : G×M (g, x) → M → Tg (x) () such that Tg ◦ Th = Tg◦h , Te = I. () If G is a group, we will speak of an invertible dynamical system. We are mainly interested in discrete dynamical systems where G = N0 or G=Z () 187 Author's preliminary version made available with permission of the publisher, the American Mathematical Society 188 6. Dynamical systems and in continuous dynamical systems where G = R+ or G = R. () Of course this definition is quite abstract and so let us look at some examples first. Example. The prototypical example of a discrete dynamical system is an iterated map. Let f map an interval I into itself and consider Tn = f n = f ◦ f
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