Calculation of the Structure of a Shrub in the Mandelbrot Set

Abstract

We calculate the external arguments of the structure of any shrub in the Mandelbrot set. Before calculating, we revise, expand, and clarify some tools useful for this paper: harmonics, pseudoharmonics, the concept of structure, the structure of a shrub, and the ancestral route. Finally we present the main contribution of this paper, a three-step algorithm which allows us to calculate the structure of the shrub. In the first step, we use pseudoharmonics that were previously introduced by us, in order to calculate the first and last external arguments of a structural node. In the second step, starting from two general properties of the Misiurewicz points external arguments introduced here by us, we present a new method to calculate the intermediate external arguments. In the last step we introduce a third property that allows us to calculate the external arguments of the representatives of the branches emerging from the structural nodes.

Exact period-four solutions of a family of n-dimensional quadratic maps via harmonic balance and Gröbner bases

“…20,21 Most of these scenarios are related to the existence of saddle-node bifurcations. Thus, for instance, period-p windows inside chaos arise from a saddle-node of period-p orbits (SNp) and continue with a sequence of period doublings within the corresponding interval.…”