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Chaos Theory Revolution

 

At the end of the nineteenth century the great majority of physicists thought that Newtonian mechanics and Maxwell’s electromagnetism could explain all the phenomena of nature. It was the deterministic image of the physical world. The classical scheme postulated that if we could determine the positions and velocities at a certain moment of all the particles of the universe then we could calculate anything from Newton’s equation. The interactions would be given by electric and magnetic fields and by Newtonian gravity. Mathematics would do the rest.

The famous Lorenz attractor represents complexity in a dynamic system.

The arrival of Einsteinian (special and general) relativity changed our view of space and time inherited from Newton, although it was the quantum theory that definitively collapsed the deterministic vision of the universe.

At the same time, and very slowly, some results were emerging that would eventually become another revolution, another radical change in the way we see the world. One of the first to note that the classic scheme could not work was the great French mathematician Henri PoincarĂ© (1854-1912).He did not need to build a new physical theory. It remained within the framework of the classical equations and observed that for systems formed by 3 bodies in gravitational interaction Newton’s equations are not sufficient It is not that we are not smart enough to find closed solutions, but no matter how clever they may be the mathematicians of the future, they will never find them. There is simply no finite algorithm.

His research on the qualitative behavior of dynamical systems opened a new path in the mathematical study of physics. The new topological and geometric tools would reinforce the old analytical techniques. Poincaré was one of the first to obtain information on equations without the need to solve them.

His work has developed throughout the twentieth century and is still far from complete. Great mathematicians like Birkhoff, Kolmogorov, Moser, Smale, Arnold … have made important contributions. And many more. It would be impossible here to point them all out. But what is this revolution about which I speak? What does it mean for our world view?

This is a general scheme applicable to disciplines as diverse as planetary mechanics, fluid dynamics, electromagnetism, biology or even economics. The common denominator: differential equations. The result: the complexity in the behavior of the variables we consider. The objective: to understand qualitatively, without solving the equations, some aspects of the system. The problem: Numerical methods fail when it comes to long-term predictions.

This is commonly referred to as chaos theory. But what do we mean when we talk about chaos? Ordinarily it is understood that something is chaotic when there is confusion or disorder. This is not the chaos of which I speak. When it is said that the problem of n bodies is chaotic, it does not mean that it is disordered, after all the dynamic evolution is given by very orderly differential equations.

We must, therefore, give another definition of chaotic system. The ingredients that are generally considered essential for chaos are: hypersensitivity in the initial conditions (that is, if we calculate the evolution from similar but not identical conditions will give us very different results), existence of complicated structures or “strange “(They are not worth spheres, cylinders or things like that) and freedom. Especially freedom (well, the technical term is topological transitivity but it seems clear to me the other).

When I speak of freedom I mean that there are no restricted regions, that if we wait long enough the system will evolve from one region to another. Freedom also refers to the “geometry” of solutions, whose complexity grows with dimension, and which is not classified at all (there are indeed theorems that in certain cases there is no classification algorithm).


We must visualize this graph as a body of revolution rotating it around the vertical axis of the left. It represents the opposite of the previous example: the ordered behavior of a dynamic system. The surfaces obtained by rotating the figure in the indicated direction contain magnetic fields. As you can see, they are bulls, which implies bounded magnetic orbits.

But there are many more fascinating and complex things that are not included within what is called chaos. Instead of talking about chaos, it is more revealing to talk about the different creatures that can populate a system: bifurcations, stability or instability (of many types), attractors or repulsors, knots, borders, symmetries, conserved “magnitudes”, ergodicity … and many More, some of which we know, and others we can not even imagine.

In short, it is an unlimited and complex universe of which we still know little. The chaos is just one among many interesting things. I simply encourage all those who have drawn attention to the topic to browse some books. You will find will not disappoint your imagination.

The next great era of awakening of the human intellect may very well produce a method of understanding the qualitative content of equations.” Richard P. Feynman.

Further reading:

R. Abraham, JE Marsden: Foundations of Mechanics. Benjamin, Reading Mass (1992).
S. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990).
KR Meyer, GR Hall: Introduction to Hamiltonian Dynamical Systems and the N-body problem. Springer, New York (1992).

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