Abstract
This paper modifies the escape-time method, analyzes the process that the points inside a period bulb of the generalized Mandelbrot set (the generalized M set) after iterating a certain pre-period process finally enter into the periodic orbit. All the points with same pre-period form a circularity structure which shapes similar as the period bulb; as pre-period grows the circularity structure diffuses towards the period bulb boundary until crams the whole period bulb. Besides, we present a particular experimental observation of the rule of the pre-period bulb graphic of the generalized M set and its growing process and offer the theory proof.
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