Dynamics Beyond Uniform Hyperbolicity: A Global Geometric And Probabilistic Perspective Hardcover English by Christian Bonatti

New

Christian Bonatti

What is dynamics about? In broad terms, the goal of dynamics is to describe the long term evolution of systems for which an "infinitesimal" evolution rule is known. Examples and applications arise from all branches of science and technology, like physics, chemistry, economics, ecology, communications, biology, computer science, or meteorology, to mention just a few. These systems have in common the fact that each possible state may be described by a finite (or infinite) number of observable quantities, like position, velocity, temperature, concentration, population density, and the like. Thus, m the space of states (phase space) is a subset m of an euclidean space m . Usually, there are some constraints between these quantities: for instance, for ideal gases pressure times volume must be proportional to temperature. Then the space m is often a manifold, an n-dimensional surface for some n < m. For continuous time systems, the evolution rule may be a differential eq- tion: to each state x g m one associates the speed and direction in which the system is going to evolve from that state. This corresponds to a vector field x(x) in the phase space. Assuming the vector field is sufficiently regular, for instance continuously differentiable, there exists a unique curve tangent to x at every point and passing through x: we call it the orbit of x.

3540220666

9783540220664

English

Springer-Verlag Berlin and Heidelberg GmbH & Co. KG

384

Lorenzo J. Diaz

2005 ed.