Bifurcation Diagrams

Since changes in the value of the control parameter $r$ produce attractors with different periods, it is useful to plot the post-transient solutions against $r$. This is shown in the bifurcation diagram of Fig.(3.7a). For $r<1/4$ all orbits decay to the attractive fixed point at the origin. As $r$ crosses $1/4$ a new fixed point appears that is attractive for $1/4 < r < 3/4$, while the one at the origin becomes unstable. At $r=3/4$ there is a period doubling bifurcation because for $3/4<r< 0.86237...$, period-2 attractors exist. These period doubling bifurcations continue as $r$ crosses various thresholds, leading to period $4$, $8$ etc attractors: In each case, as a particular threshold is crossed, new period-$2^k$ attractive fixed points emerge, while the old period-$2^{k-1}$ fixed points become unstable. This is illustrated symbolically in Fig.(3.8) where the dotted line represents a fixed point that become unstable as a threshold is crossed, and as two new stable fixed points emerge (the two prongs of the pitchfork) . Finally at $r=r_c=0.8924865....$ an infinite period results (strange attractor) leading to chaos.

As $r$ is increased beyond $r_c$, ‘windows’ sometimes open up in the diagram as periodic attractors recur at some points. Notably, now one begins to see periodic attractors at odd integers. As $r$ is further increased, these odd periodic attractors themselves undergo period doubling bifurcations to chaos. Indeed, it has been proven (Yorke) that period three orbits imply chaos. For the logistic map, there is a period three window near $0.9571.. < r < 0.9603..$. One feature of the bifurcation diagram that should be noted is its self-similarity. Fig.(3.7b) shows a magnified view of the boxed region in Fig(3.7a): It looks very much like the original whole. In fact such approximate self-similarity continues as further bifurcations take place. The self-similarity is also important in establishing some universality properties of the bifurcation process, as discussed below. The above discussion has been for the critical values of the control parameter. To find the period $2^k$ fixed points, $x^{*}$, one needs to solve the equation

$$x_n = x_{n+k}.$$ (3.12)