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# One-Dimensional Dynamical Systems

## Part 1: Introduction

#### Qualitative behavior

We want to understand the long term behavior of a dynamical system. In other words, we want to know how many squirrels there are after many, many years. We can find out by applying the function

f(x) = 3x

over and over again. In fact, we consider the orbit and see whether it has a limit. Does it matter whether we start with 8, 4, or 20 squirrels?

Let us start with 12 squirrels; click on the dots to see more iterates:

Starting with 4 and 20, we get:

The orbits for 4 and 12 only differ in the beginning. However, is the orbit of 20 so much different? If we go on, the number of squirrels will just become outrageously high, whether we start with 4 or with 20 squirrels. Qualitatively, the orbits are the same, the eventual behavior is the same. In fact, for any initial number of squirrels (okay, let us take at least two), we eventually end up with infinitely many. If we only look at the abstract model and forget about biological complications, one squirrel is also enough to end up with infinitely many. However, if there are no squirrels at all this year, there won't be any ever.

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Written by Hinke Osinga