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# One-Dimensional Dynamical Systems

## Part 1: Introduction

#### Exponential growth model

Consider the number of squirrels on the university campus. Let us try
to come up with a model for the population growth of these
squirrels. If we think in averaged numbers, we may assume that the
growth rate is constant over the years. Consider a simple case where
each couple has 4 babies each year and no squirrels die. This means
that we have 2 newborns per squirrel. The growth rate per year is 2.
Suppose there are 12 squirrels on campus this year. How many squirrels
will there be next year? What if this year there are only 4, or 20?
Depending on the number of squirrels this year, say *x*, the
number of squirrels for next year can be expressed as a function:

*f*(*x*) = *x* + 2*x* = 3*x*.
If the number of squirrels for this year is 12, then we will have
*f*(12) = 36 squirrels next year. The year after that, there will
be *f*(36) = 108 squirrels. This answer is found by applying the
function twice. How many squirrels will there be in four years? What
is different if we start with 4 squirrels this year? The repeated
application of a function is called iteration. Here is a formal
definition.

**Definition (Iteration):** For a function *f* and a point
*x*, *f*(*x*) is called the *first iterate*
of *x*, and *f*(*f*(*x*)) is called
the *second iterate* of *x*. Repeatedly evaluating the
function like this is called *iteration*.

**Definition (Orbit):** The set of all iterates is called the
*orbit* of *x*.

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Written by Hinke Osinga

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Created: Mar 31 1998 ---
Last modified: Wed Apr 8 16:56:23 1998