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Redalyc.FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND Investigación Administrativa ISSN: 1870-6614 [email protected] Escuela Superior de Comercio y Administración, Unidad Santo Tomás México Barnsley, Michael; Ramos, María; Villasante, Sebastian FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION Investigación Administrativa, núm. 110, julio-diciembre, 2012, pp. 49-56 Escuela Superior de Comercio y Administración, Unidad Santo Tomás Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=456045338004 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative FUNCIONES DE LA ECONOMÍA FRACTAL: COGNICIÓN Y PARTICIPACIÓN FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION Michael Barnsley (1) María Ramos (2) Sebastian Villasante (3) ABSTRACT In this paper we present fractal modeling that will help us understand and analyze the prospects of reaching the stations listed its shares in the financial market and the correlation of the latter with the company, including relaxation of the ranges we purchase and sales functions sine and cosine to serve as basis for investment circles via fractal Fibonacci series is non-parametric statistics with golden mean. Key words: Fractal, iteration investment circle, alternating parameter, smoothing. RESUMEN En este trabajo se presenta el modelado fractal que nos ayudará a comprender y analizar las 49 perspectivas de alcanzar prospecciones de las acciones en el mercado financiero y la correlación de ésta con la empresa, incluyendo la relajación de los rangos de nosotros compra y venta de seno y coseno funciones para servir como base para la inversión a través de los círculos fractal serie de Fibonacci es estadística no paramétrica con media dorada. Palabras clave: Fractal, iteración de un círculo de inversión, alternancia paramétrica, suavización. Clasificación JEL: C6: Métodos matemáticos y programacion y C14: Métodos semiparamétricos y no paramétricos. (1) BA Matemáticas – Oxford University - 1968, Ph.D Teórica Química – University of Wisconsin - 1972. Profesor de Matemáticas, Universidad Nacional de Australia - 2004. [email protected] (2) Lic. Economía - UNILA - 2000, Master Finanzas- Universidad de Londres -2008, Ph D Economía - Instituto Politécnico Nacional con estancia en Universidad de Santiago de Compostela - 2012. [email protected] (3) Lic. Derecho - Universidad de Buenos Aires - 2000. Master Economía - Universidad de Santiago de Compostela - 2005. Ph D Economía - Universidad de Santiago de Compostela con estancia en The Beijer International Institute of Ecological Economics - 2009. [email protected] #110 / Año 41 Barnsley M.; Ramos M.; Villasante S. FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION INTRODUCTION ¾. In practical terms, a fractal is a structure that is composed of small parts, which are similar to the It is vital for all concerned whit capital market original figure, repeated in different scales, from alternatives such as Dimensional Fractal to large (macro) to small (micro) [Alligood, K.Yorke, expand our knowledge on them. One first step J: 1996]. The fractal approach states that the path would be to describe different existing theoretical from the microcosm to the macrocosm is similar models because this would allow to compare and constitutes a new field of mathematics and is while we establish the relevant the securities involved in changing the paradigms of science, in markets that are related in terms of theory our case of the financial economy. The nature Securities mathematical activity, as in the case of expresses itself through its own geometry: Frankfurt (DAX30), London (TSE), Paris (CAC- Fractals, which is interesting for the analysis of 40), Tokyo (Nikkie-225) and New York various processes in both nature and society (STANDARD & POOR'S) for Mexico (IPyC). To [Shoroeder, F: 2003]. The dynamics of a fractal appreciate the value and diversity of the can lead to a more complicated behavior. For Dimensional Fractal approach, we believe that such systems, small differences in initial the spread of chaos theory and fractal geometry conditions can grow in large changes in stock in the social sciences is essential to our future values ?in markets, where we defines the following evolution, as the art of storytelling was essential recursive propositions referring to stock prices, the origins of our culture [Mandelbrot, B: 1999]. which are built based on basic financial data: There is no rigorous definition with mathematical precision that defines whether a given set or not a Y=f(x)…………………………………………....(1) fractal. However, most authors agree that Fractal is considered a final product that originates Role of Participation, relative to market through the iteration of a geometric process. This performance: is usually very simple in nature and originates in the successive iterations sets of certain size, X=f(y)………………………………………….(1.1) fixed throughout the process that is modified by converting the infinite iteration. According to this theory, these two propositions depend on each other and feed each other, and 50 This paper is organized as follows. In Section 1 that stock prices vary depending on the we contextualize the international Capital information available on the market. This is the Markets according to their type and logistics sensitivity to initial conditions4 (Ex post price) and fractal (mapped and parameterize prices) it represents one of the attributes of chaos. The showing the effects of the non-linearity, reflexivity pattern is identical to that produced by a random and the collection of attractors in market functions IFS with certain pairs of prices excluded. Others (cognitive and participation). This Section also require more excluded combinations (triples, considers the price taps geo-fractal iterations for quadruples, etc.). The number of combinations is training in Sierpinsky Forks, Cantor and Julia. excluded from this measure of complexity. In Section 2 we show the geometry of nature in Figure 1: Map of Fractal logistics the stock prices, its effect on the topological dimension of study considering the golden mean and the exponent of Hurts. Finally, Section 3 presents the conclusions and recommendations for general use our method. Definition, Types and Logistics Fractal a) Long memory b) Short memory Capital Market The test functions for our economic and financial The term “Fractal comes from the Latin “fractus” dynamics research are the logistic map and the meaning “fragmented”, “broken” or simply “torn or broken”. The term is applied to all forms normally generated by a process of repetition wich are (4) The sensitivity to initial conditions implies that there is exponential divergence of trajectories initially very close in three- characterized by full-scale similarity, not to be dimensional space, a phenomenon known as stretched (change differentiable and to exhibit fractional dimension in prices for finance). INVESTIGACIÓN ADMINISTRATIVA AÑO: 41 NUM. 110 ISSN: 1870-6614 Julio - Diciembre 2012 Barnsley M.; Ramos M.; Villasante S. FRACTAL ECONOMY FUNCTIONS: COGNITIVE AND PARTICIPATION map of the store. The logistic map is defined by a Figure 1.2: Iteration Fractals parabola, the map of the store by a broken line [Diebold, F. Nason J: 1990], both symmetric about x = ½. The height parameter gives the total financial activity, which in our paper is the number of stations that exist in the equity capital markets. Figure 1.1: Parameters of a Fractal The loss of predictability is the bad news of the chaos in financial economics, while the good news is a much more robust control of chaotic a) Logarithmic b) Exponential c) Three-dimensional systems in the stock market. Under occasional slight changes in the price of stock system, it is Breaking the map into a discontinuous function possible to stabilize any of unstable periodic the bending of the graph becomes a bifurcation orbits pervading chaos. Each point of the orbit is diagram. in a drawer (or time), and as it follows in the orbit (market trend) it increases each point of the The left side of the diagram resembles the leaf horizontal line. Then, it is possible to determine tent (Gaussian), but the left side shows a an approximate measure of the amount of time sequence of periodic windows wich are keeping that passes into the orbit of each price interwoven in the same order as the disks region5. The starting point is an original connected all the cardioid of the Mandelbrot set. information which is then processed to obtain the Iteration graph of a fractal orbit is produced by result. This is processed again (iterates) and one generating points (x0, x), (x1, x2), (x2, x3),... in a obtains another similar result to the above and three-dimensional system: continues doing the same procedure indefinitely with each outcome. The Hausdorff-Besicovitch 51 (x, y, z) …………………………………………(1.2) dimension6 is: Where: S=LD………………...................................(1.2.1) x = Maximum price range y = Minimum price range Where S is the number of segments or their z = Logarithm of shares outstanding length, L is the measuring scale; D is exactly the size, then we obtain: From x = x0, with a vertical line on the graph y = f (x) from which we repeat function are cut into (x0, LogS=LogLD……………….......................(1.2.2) f (x)) = (x0, x1), and return to trace the horizontal line (defining market
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