Dynamical systems are collections of states which evolve over time in some prescribed way.  The states can be very simple (position of a particle moving back and forth along a line) or horrendously complicated (weather systems, economic systems, ecological systems, systems of brain architecture and activity).  Most of the horrendously complicated ones have thus far defied attempts to develop useful models predicting future states from current ones. This raises the question of whether such systems are inherently "chaotic", which, among other things, means so sensitive to random perturbations that the tiniest disturbance may drastically alter the future.  It's often said that weather systems seem to be so unstable that the decision of a bird to dive after a worm in Southeast Asia may determine whether or not a tornado develops in the Caribbean.

           Somewhat surprisingly, even fairly simple mathematical models often lead to chaos in the sense mentioned above.  In trying to develop a model for a system of only moderate complexity, it's therefore very important to have a feeling for what sorts of models may produce chaos and, vice versa, what sorts of models predict that no matter what the current state, the system will eventually settle down to an equilibrium state with random perturbations not effecting the equilibrium state.  It was thought for quite a while that planetary systems always settle into equilibrium (planets going around the sun and moons going around planets in fixed orbits, comets and asteroids following predictable paths,...).  We now know this isn't true.

           The development of a systems model is a long slow business.  In 312, we won't discuss the highly sophisticated tools in statistics and fancy computer packages needed to organize large data sets and begin to look for patterns. What we will do is look at mathematical models of various sorts which might be pertinent to a modeling team after the expenditure of many millions of dollars and thousands of computing hours to detect some patterns.  Our emphasis will be on examples, especially those with simple mathematical formulations which initially appear too "simple-minded" to be of any use for complicated real-life systems but which, on deeper inspection, reveal amazingly intricate patterns and/or very chaotic behaviour.  Only though such examples does one begin to get a sense of what chaos is all about.

                           Linear algebra is the single most useful tool needed in a course like 312.  Hence, a prior course like Math 309 is a necessity.  Although parts of Math 217 and Math 318 are highly relevant, these aren't crucial.  A good working understanding of the last part of Calculus 2 (Chapters 7 and 8 in Stewart's text) and the first half of Calculus 3 (especially Chapter 9 and the first half of Chapter 11) will be far more useful than a hazy grasp of courses like 217 and 318.

            We will try to cover Chapters 1-9 in the textbook, averaging about 5 class periods per chapter.  Chapters 1-6 are concerned with discrete time models where we only measure systems at integer multiples of a fixed time unit (the time unit could range anywhere from 1 millisecond to 10 million years) and assume there is a fixed transformation f taking states at time n to states at time n+1 for any n.  We'll see that chaos is present even for very simple f's.  In Chapters 7-9, we'll look at continuous time models where states evolve via a system of differential equations.  As we'll see, one can always convert from a continuous time model to a discrete time model but not necessarily so in the opposite direction.  One can therefore think of continuous time models as very special types of discrete time models, so special that it's a bit unlikely they will ever be of much use outside of certain physical science situations. For all other applications (business, social sciences, biological and behavioural sciences, all the remaining physical science situations), discrete time models are more germane.

            As mentioned above, most of our time will be devoted to working out examples. Students will be asked to volunteer to present their solutions to selected homework problems.