The top part of the picture corresponds to a single attracting fixed point of f for -3/4 < c < 1/4. For c > 1/4 points go away to +Infinity (see tangent bifurcation). Filaments and broadening show how the critical orbit points are attracted to the fixed point. At c ~ -3/4 we see a branching point due to period doubling bifurcation. Then all the Feigenbaum's cascade of bifurcations. At the lower part of the bifurcation diagram you see chaotic bands and white narrow holes of windows of periodic dynamics. The lowest and biggest one corresponds to period-3 window (there are 3 junction points in it).
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Fig.1 shows that caustics in distribution of points of chaotic orbits are
generated by an extremum of a mapping. 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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