Some of the simplest dynamical systems consist of a point moving along the number line according to a rule. For example, if we can the rule to be " given a number, reduce the number by one-half". Then if we start with an initial value of 8, after one application of the rule we get 4. After another application, we get 2, then 1, then 1/2 , 1/4, 1/8 and so on.
A key goal of dynamical systems is to determine how points move as the rule is repeated over and over again. In this example, we see that the numbers keep getting smaller and they will get closer and closer to 0. Note that the values never get exactly to 0; they are always a tiny bit bigger than 0. We say that the limit of the values is 0. The concept of limit is very important in mathematics.
If you play the iteration game on a calculator, after iterating the initial value 8 many times, the calculator may show the value to be exactly 0. A calculator can only keep track of a fixed number of decimal places (often 8 or 16 depending on the calculator) so when a number gets too small, the calculator thinks it is 0 when it is really not. Keep this problem in mind when you play the iteration game.
Suppose we start with a different initial value (20 or -10 for example). What values do we get as we apply the rule and where do the points move towards?
The Iiteration Game allows us to study this question. You can choose from one of several different rules, enter your own initial value and see its iterates. The rule is written in the form of a function f(x). For example, the halving rule would be written as f(x) = 1/2 x. Among the rules you can study are the doubling rule ( f(x) = 2x), the square root rule ( f(x) = sqrt(x)), the squaring rule ( f(x) = x^2), the sine rule ( f(x) = sin x), the cosine rule ( f(x) = cos x) , and the logistic rule f(x) = 4x(1-x).
The iteration game displays the iterates as numbers. Sometimes it is easier to understand the dynamics when we look at a picture. The Staircase Game provides a pictorial way to look at iteration. Discussion of Mathematics.
The game is set up to study the dynamics of the logistic family in which the rule of motion is f(x) = kx(1-x). For each k value, we get a slightly different rule of motion. Changing the k value is like changing the shape of a billiard table; it will cause the behaviour of iterates to change. We would like to see what happens for different k values.
One can choose k values between 0 and 4. You can choose an initial starting point any where in the interval 0 to 1. You choose how many iterations the program will produce.
For some k values, it is not clear what pattern the orbit is producing. The first few iterates move around seemingly without a pattern and it is only the later iterates that follow a clear pattern. To help in observing this type of behaviour, you can tell the program to only start drawing the orbit after a certain number of iterates have already happened.
If a point does not move under the rule of motion, we call it a fixed point. For example, with the halving rule, if we start at 0 and take half of that, we still get 0. So 0 is a fixed point of the motion. Expressed in terms of the function f, a fixed point has the property that f(x) = x.
There are two main types of fixed points: attracting and repelling. A fixed point is attracting if all points that start near the fixed point approach the fixed point. You can check that 0 is an attracting fixed point for the halving rule: all that start near 0 will approach it. A fixed point is repelling if points that start near it move away from it.
As you play with the different rules, see if you can determine the fixed points and decide if they are attracting or repelling.
f(x) = 1/2 x. The point x=0 is a fixed point and it is attracting. All other initial values will approach 0.
f(x) = 2x. The fixed point is x=0. This fixed point is repelling; nearby points move away from 0. Any positive initial value will just get bigger and bigger; it moves off towards positive infinity. Any negative initial value will get bigger and bigger negative and move off towards negative infinity. Since these numbers are moving to the left along the number line, we say they are getting smaller - even though the size of the number is getting bigger! A confusing situation.
f(x) = sqrt(x). Since the square root is not defined for negative numbers. There are two fixed points: 0 (repelling) and 1 (attracting). Any initial value in the interval (0,1) will approach 1. Initial values bigger than 1 will all approach 1. This example is tricky. If you start with a value bigger than 1, it will get smaller with each application of the rule. You might predict that since the values keep getting smaller, they will approach 0; this is not the case! They approach but never reach 1.
f(x) = x^2. The fixed points are x=0 and x=1. The fixed point x=0 is attracting; any point that starts in the interval (-1, 1) will approach 0. The fixed point x=1 is repelling; if x is in the interval (0, 1), it goes towards 0 and away from 1. If the initial value is bigger than 1, the iterates go towards positive infinity. Note that a negative initial value becomes positive and then stays positive.
f(x) = sin(x). A word of warning. This function is defined in terms of radians not degrees. The point x=0 is an attracting fixed point. All other initial points will iterate towards 0. However, they will approach 0 very, very slowly!
f(x) = cos(x). Any initial point will iterate towards the value x=.739. This point is an attracting fixed point for the map. It is not easy to figure out which point solves the fixed point equation f(x) = x.
f(x) = 4x(1-x). CHAOS! If we start with an initial value x in the interval (0,1), the orbit will not have any fixed pattern. The numbers will keep varying and never approach any fixed value. This is an example of chaos. There are a few points whose orbit does have a predictable behaviour. For example, x=0 is a fixed point. And x=1 will go to x=0 and then stay there. Can you find other initial values that have simple orbits?