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# MODULE: Dynamical Systems

## Part 1: Introduction.

Dynamical systems is the study of systems governed by a consistent set of laws over time. These systems can model an enormous range of behavior, such as the way in which your coffee cools while you drink it, the amount of interest your money is earning in the bank, the rate at which the world's human population increases, and the motion of the planets in the solar system. We are interested in understanding the long term behavior, or dynamics, of these systems. In this lab, you are going to perform a computer-assisted study of a simple system which is related to a model for population growth.

Even very simple dynamical systems, such as the one in this lab, can result in highly complicated behavior. Most of the time, it is impossible to give exact numerical values at which behavior occurs in a dynamical system. We can only study the qualitative behavior. Thus the qualitative approach, such as you will take by using a computer to study behavior, is the key to understanding the material.

#### Mathematically Speaking

It is all very well to say we are embarking on the study of systems governed by a consistent set of laws over time, but mathematically, how can we interpret this? By a set of laws, we just mean a function. To say that this is consistent over time, we mean that we apply the same function over and over again. For example, suppose your function is the square root function; using a scientific calculator, type in any positive number. Take the square root of that number. Take the square root of the result. Take the square root again. And so on.

Now you know what a dynamical system is, but what are we trying to do with them? We said before that we are interested in understanding the long term behavior of these systems. Mathematically, this means we want to answer this question: if we are given a number, and we apply our function to it over and over, what happens? Of course, this may depend on which number we are given. For the square root function you used above, you may have noticed that eventually number displayed by your calculator stops changing. Write down the number you see. Repeat the whole process with a different starting number. Keep trying this with different numbers until you have a conjecture about the long term behavior under the square root function.

What happens if you use some function other than the square root? What sort of behavior do you get by repeatedly taking the cosine of a number? The square of a number? What about repeatedly pushing the "1/x" or "+/-" buttons?

The repeated application of a function is called iteration. Here is a formal definition.

Definition (Iteration): For a function f and a point y, f(y) is called the first iterate of y. f(f(y)) is called the second iterate of y. Repeatedly evaluating the function like this is called iteration. Definition (Orbit): The set of all iterates is called the orbit of y.

#### Why Should You Want to Understand Dynamical Systems?

So far this has all been quite abstract. How does the mathematical description of eventual behavior of points under iteration by a dynamical system relate to all the nice examples given in the first paragraph?
##### Interest
Here is how it relates to the interest on your money in the bank. If your bank tells you that your interest will be compounded annually at an annual percentage rate of r, and you put in P dollars, how much money will you have after one year? You have probably learned that the answer is given by the function F:

F(P)= (1+r) P

What about after two years? Since the bank compounds annually, we need to include in the second year, the money earned in interest in the first year. Thus the interest earned after two years is the second iterate of the function F. The amount of money earned after n years is the nth iterate of F.

##### Population Growth
The simplest model for population looks just like the model for earning interest. In other words, the increase in population depends only on the current population P and the rate of population growth r. Thus the function F is a simple model for population growth. However this model is so simplistic that it does not take into account the possibility that there may be finite resources. In other words, for an overcrowded population, the rate of growth will decrease, due to scarcity of food and other essential needs. A more sophisticated model of population takes the limited resources into account by adding a term which decreases as P becomes large. The new function looks like

G(P)= (1+r) P (1 - b P).

It would be nice to know how population changes over many years. Does it die out, does it explode, or is it something in between? In other words, we are interested in the long term behavior of points under iteration by G. In general, this depends on the variables r and b. In the following section, we study iteration under a similar set of equations. Keep the idea of population in mind and think about the implications of your findings related to such a model.

Iterated functions come up in a variety of mathematical models. In order to better understand these models, we will look at how the long term behavior changes.

#### Eventual Behavior

Here are a few more concepts so you can describe the long term behavior of iterates. As you learn these terms, you will look at the function f(x)=x^2, the squaring function. You should have a button for this on your calculator.

Definition (Fixed Point): For a function g(x), a point p is a fixed point when g(p)=p.

• What are the fixed points of f(x)=x^2?

• Pick a point "near" (distance less than .1 is near enough) a fixed point. Keep iterating (pushing the squaring button) until you see a pattern for the long term behavior. Try again with other numbers near each of the fixed points. Write down what happens for each choice. Does it matter if the number you pick is less than or greater than the fixed point?

Notice that for the squaring function, some orbits converge to a point. Some orbits diverge (get arbitrarily large). Are there orbits which always stay bounded, but do not converge to just one point? In general, the answer is yes. You will observe more complicated long term behavior of orbits in your computer investigations.

Definition (Attracting and Repelling Fixed Point): A fixed point p is easy to detect computationally when orbits of nearby points converge to p. In this case, p is called attracting. If the orbits of nearby points move away from p, p is called repelling. A fixed point which is neither attracting nor repelling is called neutral.

• Which of the fixed points for the squaring function are attracting? Which are repelling?

• Every point of the identity function i(x)=x is a neutral fixed point. Do you see why?

Next: Parametrized Families and Periodic Points