Chaotic and fractal field line calculations using decomposition and self winding

Abstract

Faster methods for topological categorization and field line calculations are developed by using decomposition regions together with the self-winding techniques first developed in a prior patent application. A point iteration technique provides direct calculation of low order digits of winding counts without use of complex intervals. Easy to calculate derivatives define decomposition interval boundaries which substitute for methods using the slower complex interval processing of the prior patent. Methods common to this and the prior patent are developed for visualizing conformal mappings of iterated functions.

1 . A machine executed winding digit adjustment calculator method to return a winding digit adjustment for a metacomplex escape topology based on a predetermined metacomplex iteration expression, a predetermined winding adjustment escape, and a predetermined precision escape distance, comprising:
 (A) providing a maximum reducer means to transform said metacomplex iteration expression into a real valued expression, wherein beginning with a initial value, each real valued iterate always has an absolute value greater than any absolute value of an iterate of the complex iteration expression started with any complex value of the same magnitude as said initial value,   (B) providing a winding adjustment iterator means to count the number of iteration steps, minus one, to iterate said real valued expression, beginning with the predetermined winding adjustment escape, until the absolute value of the iteration first achieves said predetermined precision escape distance,   (C) computing a winding digit adjustment using said winding adjustment iterator means.   
     
     
         2 . The machine executed method of  claim 1  wherein said predetermined winding adjustment escape is a global escape distance for said predetermined metacomplex iteration expression. 
     
     
         3 . The machine executed method of  claim 2  wherein said predetermined metacomplex iteration expression is a complex expression. 
     
     
         4 . A machine executed full modulo iterator method for creating a modulo point from a predetermined point iterate data, a predetermined point iterate goal, a predetermined winding digit adjustment, and a predetermined complex iteration expression, comprising:
 (A) providing a branchcut adjuster means to return a winding count adjustment for a predetermined receiver point by counting directional transversal of the negative axis branchcut arising from the shortest rotation angular adjustment to rotate from a predetermined donor point to said predetermined receiver point,   (B) providing a complex dominant splitter means for separating said predetermined complex iteration expression into a dominant expression as the term having the highest degree of an iteration variable and setting an inferior expression to the remainder of the expression,   (C) providing a self winding operator pool means for returning a winding operator from a pool of the winding operators based on a predetermined winding operator request,   (D) providing a self winding dominant stepper means for returning a winding point resulting from iterating said dominate expression in a predetermined iteration direction, using a predetermined point iterate data, by utilizing the winding operators from said self winding operator pool means,   (E) providing a modulo banding creator means to construct a modulo point from a predetermined modulo banding parameters,   (F) providing a full modulo resolver means to create a modulo point by calling said modulo banding creator means with said predetermined modulo banding parameters of (E) constructed by iterating said predetermined point iterate data to achieve said predetermined point iterate goal.   
     
     
         5 . The machine executed full modulo iterator method of  claim 4 , enhanced to eliminate self winding calls, with the new method called an optimized modulo iterator method, further including:
 (A) providing a traditional iterator means of returning a traditional result by evaluating a predetermined domain start point with said predetermined complex iteration expression using traditional evaluation of iterated functions until a predetermined escape distance is achieved,   (B) providing an optimized modulo resolver means to both call the traditional iterator means to obtain a partial iteration result called the traditional result and then to use said traditional result as part of a modified point iterate data with a winding point iterate goal constructed from the predetermined point iterate goal, as parameters to the full modulo iterator method to produce a modulo point as the final result,   
     
     
         6 . A machine executed codomain point resolver method to convert a predetermined modulo point into a receiver codomain point by setting the field band integer of said predetermined modulo point based on a predetermined donor codomain point, comprising:
 (A) providing a codomain same level resolver means to set said field band integer based on disambiguating the enclosing decomposition region of said predetermined modulo point based on input of the field band fraction of said modulo point and the donor field band from said predetermined donor codomain point,   (B) optionally providing a codomain up level resolver means to set the receiver field band integer by calling said codomain same level resolver means on the field band fraction of said predetermined modulo point and the field band of said donor codomain point after adjusting the field band of said donor codomain point to appear to have a higher escape iteration count which matches the escape iteration count of said predetermined modulo point,   (C) optionally providing a codomain down level resolver means to set the receiver field band integer by calling said codomain same level resolver means on the field band fraction of said predetermined modulo point and the field band of said donor codomain point after adjusting the field band of said donor codomain point to appear to have a lower escape iteration count which matches the escape iteration count of said predetermined modulo point,   (D) providing a codomain resolve chooser means which chooses a means from those available in (A), (B), or (C) based on comparing escape iteration counts of said predetermined modulo point and said predetermined donor codomain point.   
     
     
         7 . The machine executed codomain point resolver method of  claim 6  wherein at least one of the optionally provided means of said codomain up level resolver means of (B) and said codomain down level resolver means of (C), is available for said codomain resolve chooser means of (D). 
     
     
         8 . The machine executed codomain point resolver method of  claim 6 , wherein both of the optionally provided means of said codomain up level resolver means of (B) and said codomain down level resolver means of (C), are present and useable by said codomain resolve chooser means of (D) 
     
     
         9 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 1 .   
     
     
         10 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 2 .   
     
     
         11 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 3 .   
     
     
         12 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 4 .   
     
     
         13 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 5 .   
     
     
         14 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 6 .   
     
     
         15 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 7 .   
     
     
         16 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 8 .