If f is a function from a certain set A to the same set A, then for any x in A, f(x) is also an element of A. Consequently, it makes sense to apply f to the element f(x). If we do this, we obtain another element of A, which could be denoted by f(f(x)). Similarly, we could apply f to f(f(x)), to obtain f(f(f(x))). If we continue this process, we will obtain the sequence: x, f(x), f(f(x)), f(f(f(x))), ... . This sequence is called the orbit of x under f.
As a notational convenience, we denote f(f(x)) by f 2(x), f(f(f(x))) by f 3(x), and so on. For example, the orbit of x = .6 under is , because
If the domain A of the function f is a set of real numbers, a calculator makes it easy to compute the orbit of a number under f. For instance, suppose . To find the orbit of x = .6 under f using a TI-83, we do the following steps:
Do you think that this behavior -- convergence to 0 -- is peculiar to the starting value, .6 ? Try using several other inital values of x, and see what happens. In particular, start with x = .2, x = .99, x = -.5, and x = 1.1 , and describe the long-term behavior of the orbits. You should find that the first three of these orbits converge to 0, but the last one, the orbit of 1.1, gets larger and larger without bound. (We say that this orbit diverges.)
Attracting and Repelling Fixed Points
The number 0 is called a fixed point for the function , because f "fixes" 0; that is, f(0) = 0. In general, a fixed point for the function f is an element x for which f(x) = x. For a function that maps real numbers to real numbers, a fixed point can be interpreted as a value of x at which the graph of f and the graph of y = x intersect. (Does the function f(x) = x2 have any other fixed points?)
A fixed point may be an attracting fixed point, meaning that the orbits of nearby points converge -- are attracted to -- the fixed point; or it may be a repelling fixed point, meaning that the orbits of nearby points move away from the fixed point. Other fixed points may be neither attracting nor repelling. For , 0 is an attracting fixed point while 1 is a repelling fixed point. (Calculate the orbits of x = .999 and x = 1.001 to convince yourself.)
Using the Fixed Point Method
If a fixed point is attracting, that very fact gives us a way to find an approximate value of that fixed point -- simply start with any guess of the fixed point's value, and compute some of the terms in the orbit. Now of course there is no need to do this to find the attracting fixed point of , since it is clear that it is 0. But in other situations, this method can be quite useful.
For example, consider the problem of solving the equation cos x = x. This equation cannot be solved using algebraic techniques, since cos x is not an algebraic expression. However, we can view this problem as one of finding a fixed point for the function f(x) = cos x, and the fixed point (the value of x for which cos x = x) can be found by making an initial guess, and computing the orbit of that initial guess under the function f(x) = cos x . To do this, first be sure that your calculator is in radian mode. Then make a guess, say x = 1, and compute terms in the orbit of x as described above. After about 60 terms in the orbit, there is no change in successive terms (at least in the 10 decimal places displayed by the TI-83). The solution to the equation f(x) = cos x is thus seen to be approximately .7390851332. The neat thing about this is that any starting value will work just as well, because all orbits are attracted to this fixed point!
Isaac Newton, one of the co-creators of calculus, discovered a very general method of approximately solving equations, which is based on fixed points. This technique is now known as Newton's method. As a simple example, suppose we wish to solve the equation x2 - 2 = 0. (Now of course we know that the solution is . But the point is that Newton's method can be applied to much more difficult equations.) Basically, Newton's method involves finding an attracting fixed point for a function derived from the equation to be solved. The specific function that is to be used is where is the derivative of the function f, a construction studied in calculus. For the case of the equation x2 - 2 = 0 , the function f is f(x) = x2 - 2, and the function g that must be iterated to solve the equation f(x) = 0 is
Try it Again
This time, let's solve the equation x3 + x + 1 = 0, using x = 2 as the initial guess. The function to be iterated is
What Else are Orbits Good For?
Orbits and their fixed points are studied in the field of mathematics known as dynamical systems. One important application is to population modeling, in which successive values of a function f are thought of as the sizes of a given population in successive years. If the function f has an attracting fixed point, then a population modeled by f would have an eventual, stable size as indicated by that fixed point. In a later vignette, we will look at some aspects of population models of this type. If we take a slightly different point of view when looking at orbits, our work gives rise to a certain type of fractal sets, known as Julia sets.