One-Dimensional Dynamical Systems
Part 1: Introduction
A model that determines the evolution of a system given only the
intial state is called a dynamical system. A lot of things can be
described with a dynamical system. For example, the amount of interest
your money is earning in the bank, or the growth of the world's human
population. One should also think of systems like the weather, the sun
and the planets, chemical reactions, or electronic circuits. Even
though these are very different phenomena, they can all be modeled as
a system governed by a consistent set of laws that determine the
evolution over time. We are interested in understanding the long term
behaviour of these systems for arbitrary initial states. In other
words, we want to investigate the dynamics of the systems. We
make this more precise with a model of population growth for squirrels
on the university campus.
The exponential growth model for the squirrels is rather simple, maybe too simple. It is an example of a linear dynamical system. Linear dynamical systems play an important role in dynamical systems theory. You will find out everything about them in the Warm Up Questions. Next, we change the linear squirrel model so that it becomes more realistic (nonlinear). You will see that even simple dynamical systems, such as the nonlinear model for population growth, can result in highly complicated behaviour. This behaviour can be so complicated that in mathematical terms we say it is chaotic.