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# 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
behavior 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.
### Example

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 behavior. This behavior can be so complicated that in
mathematical terms we say it is chaotic.

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Created: Mar 31 1998 ---
Last modified: Wed Apr 8 16:55:05 1998