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
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  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.
 J.Hanssen, W.Wilcox
Lyapunov Exponents for the
Intermittent Transition to Chaos
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updated 14 July 2006