Cantor strange repeller
You see in Fig.1 that for c < -2 interval (BC) is mapped
outside invariant interval Ic and all points go eventually
to infinity. Two intervals [AB] and [BC] are mapped onto
Ic. So similar to the tent map
Cantor strange repeller with zero measure appears in quadratic maps
(you see below such repeller for c = -3 on complex dynamical plane).
Further you see this repeller on (x, c) plane
(for different c).
"Period three implies chaos"
In 1975 T.Y.Li and J.A.Yorke published the famous "Period three implies
chaos" paper. It turns out that nonlinear 1D map with period-3 orbit has
continuum of chaotic orbits. Let a,b,c make period-3 cycle
f(a) = b, f(b) = c, f(c) = a.
It follows from the Fig.2 that inverse function f(x)-1
is multivalued in (b,c). When iterated f(x)-1
value gets in this interval we can chouse any branch at random and make
Moreover if a map has period-3 orbit then it has orbits with every period.
It is particular case of Sharkovskii's theorem that a map with period-n
orbit has orbits with all periods n' preceding n in the list
1 < 2 < 22 < 23 < ...
< 227 < 225 < 223 < ...
< 2·7 < 2·5 < 2·3 < ...
< 7 < 5 < 3
where ... 7 5 3 are odd numbers.
Stable and unstable period-p cycles for quadratic map appear after
tangent bifurcation of the fop(x) map. With decreasing
c stable cycle loses its stability and two period-2p
cycles appear. Unstable cycles never die. Therefore after period doubling
cascade completion quadratic maps have infinite set of unstable periodic
Cantor strange repeller in regular
Therefore we meet complicated Cantor-like structures for c = -1.7542
corresponding to period-3 window of regular dynamics. For almost all
x in interval Ic points are attracted to
period-3 orbits (these points lie in circles). All the rest points
(after cutting these circles) make Cantor strange repeller with
zero measure. It includes unstable periodic orbits and chaotic continuum.
This regular map (with attracting period-3 orbit) is unpredictable in
some way. The map fco3 has 3 attracting fixed
points. In the picture J(0)-midgets attracted to the same fixed point
are colored in the same color. There is a funny "traffic lights" rule:
in any interval, between two biggest midgets with different colors one more
biggest circle has the third color (red, blue and green in the picture).
Therefore all colors are dense in the Ic. You see, that
basins of attraction of different fixed points are tightly interwoven again.
So the regular map has sensitive dependence on initial conditions
as since you can find two arbitrary close points which diverge under
iterations (asymptotically they go to the same cycle but with different
The basic dichotomy for real
For almost every c in [-2, 1/4], the quadratic map
fc : x → x2 + c is either regular
or stochastic 
For quadratic maps it is proven that the set of c values for which
attractor is chaotic has positive Lebesque measure and attracting periodic
orbits are dense in the set. I.e. between any two chaotic parameter values
there is always a periodic interval.
"Fat" Cantor sets
We will get a general Cantor set if in the "1/3 cutting" process we cut the
central 1/3 piece, then i.g. 1/9, then 1/27, etc.
Resulting set is topologically equivalent to the standard Cantor set, but as
since holes decrease in size very fast therefore the "fat" Cantor set has
positive Lebesque measure and fractal dimension 1 .
In the real interval -2 < c <1/4 , regions with chaotic dynamics
have nonzero Lebesgue measure and make a "fat" Cantor set. You can see below
that regular dynamics regions (black M-midgets) are dense on complex plane
along the real axis.
Amazingly regular dynamics windows (marked by m3,5,6...) for quartic maps
(the lower picture) are ordered in the same way.
You can test density and periods of M-midgets by animated
 Mikhail Lyubich
Quadratic Family as a Qualitatively Solvable Model of Chaos
Notices of the AMS, 47, 1042-1052 (2000)
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Next: "Transient" chaos
updated 8 Nov 06