Description: A discrete process is the same as a recurrence relation, e.g., the Fibonacci relation. The behavior of such a process is similar to that of a differential equation. One key difference is that solutions are sequences of numbers or vectors. Moreover, it is clear that given an initial condition one obtains a solution. As with differential equations it is interesting to study the long time behavior of solutions. This can be quite difficult even for linear systems and uses just about everything one might learn in linear algebra.
Some of the main topics include: Liapunov's direct method for testing stability of dynamical systems. How to solve linear systems in effective ways. Equivalence of linear higher order equations and linear systems. Algorithms for deciding when matrices are similar.
Text: J. P. LaSalle, The Stability and Control of Discrete Processes, Applied Mathematical Sciences 62, 1986, Springer-Verlag. The text is a first edition and used copies of the text are much cheaper than new copies. Unfortunately the electronic version of the text is not free.
Prerequisites: It is recommended that students have taken linear algebra and analysis.