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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.

Next: Warm Up Questions