Up: Introduction

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 + 2x = 3x.

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