Chaos and fractals belong to the field of Dynamics, a branch of Physics that began with Newton’s discoveries about differential equations, the Laws of Motion and Universal Gravitation. These discoveries solved the problem of two bodies interacting with gravity, but his interest in the motion of the Moon remained unanswered. Since then and for generations, mathematicians and physicists considered the ancient Greek three-body problem impossible.

Henri Poincaré, at the end of the 19th century, theorized about chance and randomness. In his studies he had experimented with non-linear mathematical systems, and exposed from a new perspective the supposed perpetual stability of the Solar System. Poincaré was the first to think of Chaos (from the Greek Kháos; unpredictable), relating the phenomenon directly to the initial conditions that generate it, being systems sensibly dependent on these. He also noted the existence of innumerable phenomena that, without being completely random, showed enormous changes in the result when their initial conditions were subjected to small changes, since they did not respond with linear dynamics.

Later, Edward Lorenz in the decade of the 60’s, in his work to predict atmospheric variations, and with the new technology of computers and the development of new hypotheses on the behavior of non-linear systems, manages to see graphically the behavior and the result of his equations, and discovers that small changes in the initial conditions of the system gave rise to enormous and unpredictable differences in the result, so that accurate predictions in the medium or long term would be impossible. Thus he discovered the so-called «Butterfly Effect», according to which the gentle fluttering of a butterfly can influence the weather in such a way that it can cause a hurricane on the other side of the planet. A clear example of how this initial sensitivity and dependence is responsible for the appearance of chaos at any moment.

For Ilya Prigogine, who studied systems far from equilibrium within the Theory of Thermodynamics, all systems are made up of other subsystems in constant motion and proves that equilibrium is the characteristic of closed systems where there are no energy flows between them and the environment where they are. On the other hand,¡ it also shows us that by forcing systems beyond their equilibrium limits, small changes can provoke large reactions or vice versa. It is at these moments when feedback loops arise which, depending on the case, can lead the system to collapse if they are negative, or generate processes of self-organization and self-feeding if they are positive.
The initial event that leads systems to break with hetero-organization can be most of the time insignificant, and can even occur by simple chance.

Prigogine demonstrates that most of reality is neither ordered, nor stable nor balanced, but rather ebullient with change, disorder and randomness, as well as being able to generate non-random structures and orderings spontaneously. It teaches us how in the disorder, in the non-equilibrium, in the Chaos, an ordered phenomenon exists.

Chaos Theory emerged in the decade of the 90’s and specializes in the dynamic systems of Nature that obey non-linear equations, giving rise to new factors of disorder, unpredictability, and the consequent appearance of chaos.

For this new branch of science, the processes of reality are stages of chaotic states and intercalated states of order, and it focuses on establishing the circumstances and conditions that make these processes change their cycle. In this way, these theories propose a universe that develops in cycles of order and disorder, in such a way that they follow each other indefinitely; a world that does not strictly follow a predictable and determined model, but has chaotic, non-linear and indeterminate aspects.

Chaos has eliminated barriers and boundaries between disciplines by discovering that it underlies a multitude of phenomena and establishes direct connections in different fields of science: physics, biology, chemistry, ecology, computer science, robotics, medicine, economics, music, psychology, geology, engineering, astronomy; apparently completely different phenomena such as turbulence, weather, the formation of mountains or the distribution and shape of the branches of a tree, the stock market index and other behaviors are very similar in their evolutions and developments.

Fractal Geometry shows us objects, forms and shapes that have been generated by the dynamics of Chaos. And we find fractals in our own body: the self-similarity, characteristic of fractal systems is found in different anatomical systems and organs, vascular network, arteries, neuronal networks, blood vessels including the heart; from the aorta to the capillaries, placenta, bronchi, alveolar ramification in the lungs, intestinal, biliary and bronchial tubes.

The use of fractals and Chaos Theory is fundamental in the description of complex phenomena and non-linear dynamics. They appear in the movement of fire, in the smoke of a cigarette, in electrochemical deposits and aggregates, in the trajectory of dust particles suspended in the air, in the dynamics of growth of bacterial colonies, in political theory, in tornadoes, squalls, lightning, turbulence, in the dynamics of populations, in electronic and electromagnetic signals, in chemical reactions; they are in the clouds; in their shapes and movements, in air currents, sea currents and in the land we walk on.

From the geographic location of epicenters to the formation of a fault, everything responds to a fractal pattern. The rhythm of the heart has a fractal pattern; and so does the movement of our eyes, when we look to capture information.