A wide range of disciplines such as economics, physics, chemistry, and ecology deal with dynamical systems. The economic laws of supply and demand, the solar system, the motion of a a pendulum, weather prediction, chemical reactions, predator-prey interactions, and the evolution of a disease are just a few examples of real problems. In general, Dynamical Systems come to play whenever time evolution of a system is studied.
This course serves as an introduction to the mathematical theory of systems in evolution with emphasis on geometric and qualitative techniques. Mathematical formulations of concepts such as periodicity, stability, attractors and ``chaos'' is studied. The following can give you the flavor of this subject:
Suppose the points of the real line move according to the rule ``x moves to cos(x)''. We have a dynamical system if we consider iterations of this process. A point x ``jumps'' to its new location cos(x); after two jumps it is in cos(cos(x)) and so on. A fixed point is then a solution of the equation x = cos(x). Try this on a calculator! (Don't forget to set it to radians). Choose a starting number and start hitting the cosine button over and over. What do you observe?
Among the interesting questions to explore are: Fixed points stay ``fixed forever'' but, what happens to nearby points? Is the fixed point attracting or repelling neighboring points? Are there any periodic motions? Are there any points of period, for instance, 167? For general systems we may ask what are the ``good'' concepts to define. What do we understand by random motion? For what systems are we going to say that we have chaos? How predictable or unpredictable can things be?
With a mathematical approach we will show that very simple rules of
movement can in fact generate phenomenally complicated and rich
dynamics, some even giving ``Chaos''. Philosophical consequences.
Determinism. Some movements are clearly stable, small perturbations
won't change things much. However, we will see amazing phenomena people
have discovered even in the simplest dynamical systems such as ``x
moves to 
''. For some orbits, the tiniest deviation you
can imagine from an initial position will dramatically change the
orbit's fate.
Topics from recent years: