Chaotic transient

It seems very strange that quadratic maps pass from regular to chaotic dynamics by infinitesimal change of c value. But strange Cantor repeller results in chaotic transient orbits (with positive Lyapunov exponent L > 0). These orbits appear for c values corresponding to regular dynamics windows. For large n these orbits go to attracting cycle. Chaotic transient length depends strongly on starting point x_o.

c = -1.5749 corresponds to the superstable period-7 cycle. Chaotic transient length grows for more narrow windows of regular dynamics. Therefore for tiny windows (near any chaotic c value) one can distinguish regular or chaotic dynamics only for very long iterations.

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updated 8 Nov 2006