Iteration and Differentiation

 

In the previous section we introduced the equation 

 

known as the quadratic map. We write when we want to make the dependence of f on the parameter explicit.

In this section we review two mathematical tools we will need for the rest of the chapter: iteration and differentiation. We think of a map as generating a sequence of points. With the seed , define and consider the sequence , as an orbit of the map . That is, the orbit  is the sequence of points

where the nth iterate of is found by functional composition  n times,

When determining the stability of an orbit we will need to calculate the derivative of these composite functions (see section 2.5). The derivative of a composite function  evaluated at a point is written as

 

The left-hand side of equation (2.6) is a shorthand form for the right-hand side that tells us to do the following when calculating the derivative. First, construct the nth composite function of f, call it . Second, compute the derivative of . And third, as the bar notation ( ) tells us, evaluate this derivative at . For instance, if , n = 2, and , then

Notice that we suppressed the bar notation during the intermediate steps. This is common practice when the meaning is clear from context. You may sometimes see the even shorter notation for evaluating the derivative at a point as

 

which is sufficiently terse to be legitimately confusing.

An examination of the dynamics of the quadratic map provides an excellent introduction to the rich behavior that can exist in nonlinear systems. To find the itinerary of an individual orbit all we need is a pocket calculator or a computer program something like the following C  program .

 /* quadratic.c: calculate an orbit for the quadratic map

		   input:		  l 		 x0

					output:		 1		 x1

		 			         2		 x2

		  		         3		 x3

		  			         etc.

*/

#include   stdio.h  

     ¯     ¯     ¯     ¯     ¯     main()

  

		int n;

		float lambda, x_zero, x_n;

		printf("Enter: lambda x_zero  n"); scanf("%f %f", &lambda,
&x_zero);

		x_n = x_zero;

		for(n = 1; n ;SPMlt;= 100; ++n)   

				x_n = lambda * x_n * (1 - x_n);  /* the quadratic map */

				printf("%d %f  n", n, x_n);