Bifurcation Diagram

 

To explore the dynamics of the quadratic map further, we can choose an initial condition and a parameter value , and then iterate the map using the program in section 2.2 to see where the orbit goes. We would notice a few general results if we play this game long enough.

First, if and , then the graphical analysis of section 2.5.1 shows us that all points not in the unit interval will run off to infinity. Further, if , then the same type of graphical analysis shows that the dynamics of the quadratic map are simple: there is only one attracting fixed point and one repelling fixed point. These fixed points are the period one orbits calculated in the previous section.

Second, the initial condition we pick is usually not important in determining the attractor, although the value of is very important. We seem to end up with the same attractor no matter what we pick. The quadratic map usually has one, and only one, attractor, whereas most nonlinear systems can have more than one attractor [5]. The bouncing ball system, for example, can have two or more coexisting attractors.

Third, as we will show in section 2.11, almost all initial conditions run off to infinity for all . There are no attractors in this case.

Therefore, when studying the quadratic map, it will usually suffice to pick a single initial condition from the unit interval. If ever leaves the unit interval, then it will run off to infinity and never return (provided ). Further, when studying attractors we can limit our attention to values of . If then the only attractor is a stable fixed point at zero, and if there are no attractors.

For all these reasons, a bifurcation diagram  is a particularly powerful method for studying the attractors in the quadratic map. Recall that a bifurcation diagram is a plot of an asymptotic solution on the vertical axis and a control parameter on the horizontal axis. To construct a bifurcation diagram for the quadratic map only requires some simple modifications of our previous program for iterating the quadratic map. As seen below, the new algorithm consists of the following steps:

1. Set , and (almost any will do);
2. Iterate the quadratic map 200 times to remove the transient solution, and then print and for the next 200 points, which are presumably part of the attractor;
3. Increment by a small amount, and set to the last value of ;
4. Repeat steps 2 and 3 until .

A C program implementing this algorithm is as follows .

 /* bifquad.c: calculate bifurcation diagram for the quadratic map.

		   input:		 (none)

					output:		 l1 		 x200

		 			         l1 		 x201

		  		         etc.,

		  		         l2 		 x200

		 			         l2 		 x201

		  			        etc.

*/

#include ;SPMlt;stdio.h;SPMgt;

     ¯     ¯     ¯     ¯     ¯     main()

  

		int n;

		float lambda, x_n;

		x_n = 0.1;

		for(lambda = 1; lambda ;SPMlt;= 4; lambda += 0.01)   

				for(n = 0; n ;SPMlt;= 400; ++n)   

						x_n = lambda * x_n * (1 - x_n);

						if(n ;SPMgt; 199)

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

				  

		  

  

When plotted in Figure 2.14 (for ), the output of our simple program produces a bifurcation diagram of stunning complexity .

  
Figure 2.14: Bifurcation diagram for the quadratic map.

Above the diagram we provide comments on the type of attractor observed, and on the horizontal axis significant parameter values are indicated. This bifurcation diagram shows many qualitative similarities to bifurcation diagrams from the bouncing ball system (compare to Figure 1.16). Both exhibit the period doubling route to chaos. For the quadratic map an infinite number of period doublings occur for . Both also show periodic windows (white bands) within the chaotic regions. For the quadratic map a period three window begins at and a period five window begins at . Looking closely at the periodic windows we see that each branch of these periodic windows also undergoes a period doubling cascade.

Bifurcation diagrams showing only attracting solutions can be somewhat misleading. Much of the structure in the bifurcation diagram can only be understood by keeping track of both the stable attracting solutions and the unstable repelling solutions, as we did in constructing the orbit stability diagram (Figure 2.13). Just as there are stable periodic orbits and chaotic attractors, there are also unstable periodic orbits  and chaotic repellers. In Figure 2.13, for instance, we indicate the existence of an unstable period one orbit by the dashed line beginning at the first period doubling bifurcation. As we show in section 2.7.2, this unstable period one orbit is simply the continuation of the stable period one orbit that exists before this period doubling bifurcation (see Figures 2.14 and 2.22).