# 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 below 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