Transition to chaos through intermittency

Alternation of phases of regular and chaotic dynamics is called intermittency. One can see intermittency near tangent bifurcation of window of regular dynamics. 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 x passes through narrow channel. It is assumed that after every laminar phase iterations go into remote regions (where dynamic is chaotic) and then return into the regular corridor (re-injection). The first two pictures show the regular phase in which iterations diverge slowly from the point x = 0 for parameter values near the tangent bifurcation point c_• = -1.75 .

Below you see intermittent orbits with positive Lyapunov exponents. 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 (in accordance with these pictures).

Intermittent dynamics on complex plane

On complex parameter plane tangent bifurcations and intermittency take place near the cusp of every miniature M-set. For tiny M-set with period-3 intermitency takes place at Im(c) = 0, Re(c) > -1.75 . 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.

[1] J.Hanssen, W.Wilcox Lyapunov Exponents for the Intermittent Transition to Chaos

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updated 14 July 2006