This course is an introduction to the study of dynamical systems with particular emphasis on those dynamical systems that behave in a way that has been called chaotic.
At the end of the nineteenth century Poincaré becomes interested in the study of Celestial Mechanics seeking to understand more specifically the evolution of our Solar system. While the approach until then had been directed at solving differential equations of movement analytically or numerically, Poincaré proposes the use of other tools from other areas, such as Topology, Geometry, and Algebra and Analysis, in order to obtain a qualitative and, whenever possible, quantitative description of the behavior of the system. This proposal, which goes back to his thesis, marks the birth of Dynamical Systems as a mathematical subject and is focused on developing a theory capable of foreseeing the evolution of natural and human phenomena observable in various fields of knowledge.
Dynamical systems are mathematical objects used to model "some phenomena" whose state (or instantaneous description) changes over time. These models are used in physics, biology, financial and economic forecasting, environmental modeling, medical diagnosis, industrial equipment diagnosis, and a host of other applications. A dynamical system (more precisely, a discrete time dynamical system) is a way of modeling phenomena where the same law of nature acts in each period on the state of the system. In general a dynamical system is simply a map T: X -> X, where T encodes the law of nature and X represents the set of all possible states of the system. An orbit of the dynamical system T: X -> X is a sequence a, T(a), T(T(a)), ... which describes the evolution of the system over (discrete) time if it starts at time zero in the state a in X.
Sometimes the orbits starting at two initial states a and b eventually get far apart no matter how close the two states a and b are. This makes predicting the long term future of the system very difficult if the initial state a is known only approximately (as for example the state of today's weather). Such systems are called chaotic. In the course we will show that even a simple dynamical system can be chaotic. One part of the course deals with sets called fractals. These are sets which look the same under any degree of magnification. We will show how such sets arise in a natural way as the limit of an orbit of a dynamical system.
Our emphasis throughout will be on the qualitative behavior of the models, in particular, on the prediction of qualitative change in the nature of the dynamics as a system is perturbed.
We will try to introduce all this ideas through discrete one-dimensional dynamics.
More advanced details
To understand the basic ideas of dynamical systems, the nature of chaotic behavior, and to understand the geometrical and qualitative approach.
Knowledge of the definitions, terminology and techniques of analysis covered in the course. Understanding of the principles underlying the subject. Ability to understand dynamical systems studied in the course as well as new ones. Ability to express arguments and proofs clearly and accurately.
The course covers the following topics:
1. Examples and motivation. Non-rigorous introduction to ideas to be developed in the course.
2. Basic ideas and definitions related to dynamical systems: Orbits; attracting, repelling and neutral periodic points; stable sets.
3. Chaotic, expansive and expanding maps. Examples of chaotic dynamical systems. Robust chaotic dynamics.
4. Symbolic dynamics: The shift map on a symbol space.Topological conjugacy: When do two dynamical systems behave essentially the same way?
6. Neutral periodic points and generic bifurcations.
7. Bifurcations and transitions.
8. The quadratic map.
9. Fractals and Cantor sets as limits of contraction dynamical systems.
Book's Content that will be covered (not necessarily in this order; check the contents of the course)
Chapter 2. Subchapters: 2.1 and 2.2
Chapter 3. Subchapters: 3.1, 3.2, 3.3, 3.4, 3.5
Chapter 4. Subchapters: 4.1, 4.2 and 4.3
Chapter 5. Subchapters: 5.1, 5.2, 5.3, 5.4, and 5.5
Chapter 6. Subchapters: 6.1, 6.2 and 6.3
Chapter 8. Subchapters: 8.1