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For c > 1/4 quadratic map iterations diverge to infinity.
At c = 1/4 the blue parabola touches with the green diagonal and
attracting x1 and repelling x2
fixed points appear. This phenomenon is called the tangent bifurcation.
Drag blue parabola by mouse. |
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Fig.1 shows that caustics in distribution of points of chaotic orbits are
generated by an extremum of a map. Therefore singularities (painted in the
red) on the bifurcation diagram appear at images of the critical point
fc on(0).
Let us denote gn(c) = fcon(0), then
go(c) = 0, g1(c) = c, g2(c) = c2 + c, ... The curves g0,1,...,6(c) are shown in Fig.2. |
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