Transition to chaos through period doublings

There are several scenarios of transition to chaos. It is amazing that they are common for many dynamical systems.

Period doubling bifurcations cascade

As we have seen for c < c₁ = -3/4 the derivative in the left fixed point becomes less then -1. The fixed point loses its stability and an attracting period-2 orbit x_o → x₁ → x_o ... appears. The second iteration of the map f^o2(x) get two attracting points (corresponding to the stable cycle) and one unstable fixed point between them.

Note that the first picture and the central part of the second image are very similar. One need reflect in the x axis and squeeze the first image. As since f^o2(x) in the center of the right picture is quadratic-like, therefore for c < c₂ = -5/4 the attracting fixed point at x = 0 loses its stability again and we get an attracting period-4 orbit (see below) and so on. This is the period doubling bifurcations cascade.

For c₃ = -1.375 we get an attracting period 8 orbit. The central part of the image is quadratic-like again. With growth of the number of bifurcations k period of orbit n = 2 ^k becomes immensely large very quickly. Due to scaling self-similarity c_n → F = -1.401155 where F is the Myrberg-Feigenbaum point. This cascade of period doubling bifurcations leads to a very complicated chaotic behaviour of iterated points with positive Lyapunov exponent (see below). Iterations begin at x_o = F to exclude x = 0 point.

Universal scaling law

We can trace similarity of period doubling bifurcations on the bifurcation diagram of the quadratic map. You see the first bifurcation in the center and the second one at the bottom of the picture. Small image at the right bottom part of the picture is similar to the whole image.

Second image shows the second period doubling bifurcation. Again at the left bottom part of the picture we see similar squeezed image.

After the second stretching the central part of the third period doubling bifurcation coincides with the first pictures. For n → ∞ the two scaling constants converge to α = 2.5029 in the horizontal x direction (dynamical space) and δ =4.669 in the vertical c direction (parameter space).
Moreover self-similarity and these constants are universal (don't depend on detailes of map with quadratic minimum). To test the universality look at bifurcation cascade which arises for the quadratic-like map f^o3 in the biggest period-3 window.

The lower picture is stretched left bottom part of the first image.

Reverse period doubling cascade

Band merging cascade

For -2 < c < F the orbit of the map looks like a noisy cycle of periodicity p = 2^k and p grows to infinity as c goes to F. This means that the orbit is confined to p disjoint intervals which it visits in a sequential order. Thus the orbit always comes back to the same interval after p iterations (p = 2 in the picture below). On the other hand, if one looks at the points generated by f_c^o2 then the orbit stays in one interval and looks completely chaotic. As c decreases, these intervals merge in pairs so that a noisy 2^k cycle goes into a noisy 2^k-1 cycle.

Next let us watch this reverse period doubling cascade on the bifurcation diagram. Here preperiodic point m₁ = -1.54369... splits the single chaos band at the bottom of the picture into two bands (in the horizontal x direction). The second separator m₂ splits the two chaos band into four and so forth.

You see below two successive zooms of the first diagram. The three pictures begin from the superstable period-1,2,4 orbits correspondingly and end at preperiodic point m_0,1,2). The biggest windows have periodicity 3, 6, 12. It is evident that all diagrams (but not only ordinary period doubling cascade) are self-similar.

Period doubling of critical orbits

In the diagrams shown before the biggest windows of regular dynamics have period 3, 6, 12. Superstable periodic orbits corresponding to these windows undergo reverse period doubling cascade too. You see that ordinary and reverse bifurcations lead to orbits of different kind.

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updated 14 July 2006