Transition to chaos through intermittency

intermittency 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).

Intermittency for real maps

The first two pictures illustrate, how iterations diverge slowly from repelling point x = 0 in the regular phase for c near the tangent bifurcation point c* = -1.75 .

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).

Intermittent dynamics on complex plane

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.

There are fine filaments made of smaller M-sets around these M-sets too. They make Embedded Julia sets (cauliflower in the left picture) and structures with 4,8-fold symmetry.

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

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updated 23 October 2002