Real crises
Boundary crisis
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
z_{1} and z_{2} 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 < z_{2} all orbits are bounded. Conversely,
iterations of any point in x > z_{2} 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 z_{2}. For c < 2 iterations generate a
chaoticlooking orbit (a chaotic transient) until x_{n} >
z_{2}. After this happens, the orbit rapidly accelerates to
infinity. This sudden change in chaotic dynamics is called a boundary
crisis.
As the chaotic band is limited to the right by the g_{2}(c) =
f_{c}^{o2}(c) curve, one can find easy that the
intersection of g_{2}(c) = c^{2} + c and
z_{2}(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 g_{n}(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
critical orbit.
Interior crisis
A periodp window begins as c decreases through a critical
value at which a periodp tangent bifurcation creates a stable and an
unstable periodp orbits. Further the attracting
periodp 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 periodp 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 periodp.

Chaotic bands are confined by the g_{k}(c) =
f_{c}^{ ok}(0) curves (images of the critical point).
E.g. for the period3 window the central band is confined by the
g_{3}(c) and g_{6}(c). Intersection of
g_{0,3,6,9}(c) curves corresponds to the superstable
period3 critical orbit. Intersection of g_{0,6}(c)
corresponds to the superstable period6 orbit (after period doubling
bifurcation) and intersection of g_{0,9}(c) corresponds to
a new embeded period3x3 window. At last intersection of
g_{6}(c) and g_{9}(c) corresponds to the interior
crisis and preperiodic point with period3.

Here you see the central chaotic band (in red) and close to the band
(to the left) an unstable fixed point of f_{c}^{ 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.
[1] 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