Remember, that for c > 1/4, no fixed point exists, and all orbits go
asymptotically to infinity. At c = 1/4 a tangent
bifurcation occurs at which a stable and an unstable fixed points
z1 and z2 are created.
For c < 1/4 the stable fixed point undergoes period doubling
cascade followed by chaos. If c becomes less then c = -2 the
chaotic attracting orbit is destroyed and all orbits approach to infinity again.
The right picture (2D Real Mandelbrot + Julia set)
shows, that for -2 < c < 1/4 and for any initial point in the
range |x| < z2 all orbits are bounded. Conversely,
iterations of any point in |x| > z2 diverge to infinity.
Note from the Fig.1. that destruction of the chaotic orbit at c = -2
coincides with the intersection of the chaotic band with the unstable fixed
point z2. For c < -2 iterations generate a
chaotic-looking orbit (a chaotic transient) until |xn| >
z2. After this happens, the orbit rapidly accelerates to
infinity. This sudden change in chaotic dynamics is called a boundary
As the chaotic band is limited to the right by the g2(c) =
fco2(c) curve, one can find easy that the
intersection of g2(c) = c2 + c and
z2(c) takes place at c = -2. As since the critical
orbit get into repelling fixed point therefore it is a preperiodic point and
there is an intersection of all gn(c) for n >= 2
at this c value.
In a similar way on complex parameter plane every filament of the
Mandelbrot set hair ends by a preperiodic point when repeller meets the
A period-p window begins as c decreases through a critical
value at which a period-p tangent bifurcation creates a stable and an
unstable period-p orbits. Further the attracting
period-p orbit goes through a period doubling cascade to chaos,
and the attractor apparently widens into p narrow chaotic bands
through which the orbit consecutively cycles.
The window ends as c decreases through the lower critical value at
which the p points of unstable period-p orbit created at the
original tangent bifurcation first touch the edges of the p chaotic
bands of the chaotic attractor (an interior crisis).
This takes place at a preperiodic point with period-p.
Here you see the central chaotic band (in red) and close to the band
(to the left) an unstable fixed point of fc o3.
At the crisis point iterations are changed dramatically. For
c < -1.790327 orbit from narrow bands spreads to a very large
region. The last picture shows that intermittent dynamics can take place
right after the interior crisis.
Chaotic bands are confined by the gk(c) =
fc ok(0) curves (images of the critical point).
E.g. for the period-3 window the central band is confined by the
g3(c) and g6(c). Intersection of
g0,3,6,9(c) curves corresponds to the superstable
period-3 critical orbit. Intersection of g0,6(c)
corresponds to the superstable period-6 orbit (after period doubling
bifurcation) and intersection of g0,9(c) corresponds to
a new embeded period-3x3 window. At last intersection of
g6(c) and g9(c) corresponds to the interior
crisis and preperiodic point with period-3.
 C.Grebogi, E.Ott, J.A.Yorke "Chaotic Attractor in Crisis"
Phys.Rev.Lett. 48 (1982), 1507.
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updated 14 July 2002