Alternation of phases of regular and chaotic dynamics is called intermittency. One can see intermittency near tangent bifurcation of every miniature M-set (window of stability in bifurcation map of real iterations). Fig.1. shows, that for c values when an attracting point merges with repelling one and loses its stability iterations are regular and diverge slowly while z passes through narrow channel. It is assumed, that after every laminar phase iterations go into remote regions of complex plane (where dynamic is chaotic) and then return into the regular corridor (re-injection). |
Here you see the whole picture of intermittency. It's evident, that lengths
of the regular phase regions are increased, if we choose c closer to the
bifurcation point c_{*} (the second image below).
One can find [1] , that length of the regular phase is proportional to
(c_{*} - c)^{-1/2}. I.e. it is increased two times if
we decrease (c_{*} - c) four times (as it is shown in the
pictures).
On complex parameter plane tangent bifurcations and intermittency take place near the cusp of every miniature M-set. Let us examine intermittency near tiny M-set with period-3. As since iterations go to infinity for complex c in the "red" region, intermitency takes place e.g. for the Im(c) = 0, Re(c) > -1.75 ray. M-sets m7, m8 and m50 correspond to periodic orbits with periods 7, 8, 50. But there is dense set of tiny M-sets and periodic cycles along the ray. |
You see below periodic critical orbit with period 50 corresponding to
the M-set m50.
It seems quite natural, that all filaments structures (e.g. the main antennas of primary M-bulbs) can be explained by re-injection of iterations to an unstable periodic orbit. But it is amazing, why these bifurations and intermittency take place along fractal structures made of thin filaments.
[1] J.Hanssen, W.Wilcox Lyapunov Exponents for the Intermittent Transition to Chaos