Exploitation of topological categorization of chaotic and fractal functions, including field line calculations
A topological categorization method, based on inclusive intervals, provides a general method of analyzing escape topologies for discrete dynamic systems, in complex and higher dimensions, including the calculation of both potential for complex and hypercomplex and field lines for complex iterations.
1. A machine executed escape distance method to return an escape distance for a metacomplex escape topology based on a predetermined metacomplex iteration expression, having an iteration variable, throughout a metacomplex parameter space to determine an escape distance, comprising:
(A) separating, by a dominant splitter on a machine, that separates said predetermined metacomplex iteration expression into a dominant expression as the term having the highest degree of said iteration variable, and setting an inferior expression to the remainder of said predetermined expression,
(B) transforming, by a dominant reducer on the machine, said dominant expression into a real valued expression, called a real dominant expression, with absolute values taken of all factors so as to minimize the numerator and maximize the denominator,
(C) transforming, by an inferior reducer on the machine, said inferior expression into a real valued expression, called a real inferior expression, with absolute values taken of all factors so as to maximize the numerator and minimize the denominator,
(D) determining, by the machine, a predetermined real escape expression with substitution tokens called rde and rie, and with iteration token, called x, referencing the iteration variable,
(E) determining, by an escape formula replacer on the machine, a substituted real escape expression constructed by replacing the rde token and the rie tokens with said real dominant expression and said real inferior expression, respectively,
(F) determining, by a positive real root solver on the machine, the largest positive root by solving said substituted real escape expression equated to zero, for said x,
wherein each of the steps from (A) to (F) are executed as parameters become available.
2. The escape distance method of claim 1 wherein said predetermined real escape expression is a category escape expression of rde−rie−x, so that a category escape distance is returned as a result.
3. The escape distance method of claim 1 wherein said predetermined real escape expression is a precision escape expression of
rie/rde−(1-errorFraction)*relativeErrorGoal,
where said errorFraction is a fraction between greater than 0 and less than 1, and said relativeErrorGoal is a value representing a relative error goal, so that a precision escape distance is returned as a result, further including: iterating a point of said parameter space until the absolute value of said point achieves an distance equal to a result returned by calling the escape distance method.
4. The method of claim 3 further including determining, on the machine, a guaranteed precision distance as the maximum of two distances returned by calling said method of claim 3 and by calling said method using a error reduce expression errorFraction(rde−rie)−x, so that a precision escape distance is returned.