Mathematical Modeling of the Flight of the Starlings. A Particular Case of an Attractor Repeller Pairs

Araceli Giménez Lorente

doi:10.13189/ujcmj.2019.070103

Abstract

It presents a mathematical modeling of a biological system, this is a particular case of two fractal sets, an attractor repeller pair, according to the definition given by the Morse-Smalle theory; these are complementary fractal sets, since the complementary of the fractal repeller set is the attractor set, as we can see in the demonstration. The existence of this model in Nature is found in the flight of starlings. This dynamic system works like a beat of a heart, where the movement of the systole is equivalent to the contraction (attractor set) of the flight trajectory of birds and the diastole movement to its expansion (repeller set).The modeling is presented with differential equations written in Matlab code, and several images of the fractal sets associated with the flight pattern of the starlings are generated, so that we can see this model more clearly. We will begin with a definition and its subsequent mathematical demonstration to study this complex system of complementary fractals.

Highlights

The flight of the starlings follows a fractal pattern, the shape is relative to a binary system, in this case an attractor repeller pair
It presents a mathematical modeling of a biological system, this is a particular case of two fractal sets, an attractor repeller pair, according to the definition given by the Morse-Smalle theory; these are complementary fractal sets, since the complementary of the fractal repeller set is the attractor set, as we can see in the demonstration
The existence of this model in Nature is found in the flight of starlings. This dynamic system works like a beat of a heart, where the movement of the systole is equivalent to the contraction of the flight trajectory of birds and the diastole movement to its expansion.The modeling is presented with differential equations written in Matlab code, and several images of the fractal sets associated with the flight pattern of the starlings are generated, so that we can see this model more clearly

Summary

Introduction

The flight of the starlings follows a fractal pattern, the shape is relative to a binary system, in this case an attractor repeller pair. In figure 1 we can a diagram where the blue color represents the attractor and the red color is for the repeller. This diagram represents the phase change of an attractor repeller pair, so we can see the two complementary fractals at the same time. It will study the generating points of these pairs of fractals and vortices and their polarity, in relation to the lines of the magnetic field and their mathematical correlation. It is relative to the definition of the Conley theory on invariant sets of attractor repeller pair system such as tornadoes, the flight of starlings, schools of fish, solar storms, geyser system and the beating of a heart

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Discussion

Conclusion