"Intermittent" periodic orbits

In Fig.1 a few of gn(c) = fcon(0) are shown near the period-3 window. You see the same curves in the applet below.
Pattern Period-k critical orbit (and corresponding window) takes place at gk(ck) = 0 (i.e. intersection of gk(c) and the x = 0 axis which coincides with go(c) ). Points 0, ck , g2(ck), ..., gk-1(ck) form this period-k orbit.

Curves g3n(c) (n = 0,1,2,...) intersect at the periodic point c3 = -1.75488 and make a bunch in its vicininy. g3n+1(c) and g3n+2(c) make two more bunches.

Here the central g3n(c) and the left g3n+1(c) bunches are shown near the tangent bifurcation point.

Roots of g3n+1(c) = 0 and g3n+2(c) = 0 are the intersections of the left and the right bunches with x = 0. Corresponding sequences of critical orbits C(LRL)n (n = 1,2,...) with periods 3n+1 and C(LRL)nL ones with perods 3n+2 converge to the cusp of the period-3 CLR M-set. We see that there are no roots of g3n(c) = 0 near the tangent bifurcation point, therefore we have no period-3n sequence here.

You see below period-10 window and the period-10 C(LRL)3 and period-11 C(LRL)3L critical orbits. Points of "intermittent" orbits jump consequently between these three gn(c) bunches.


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updated 24 October 2002