# THE THEORY OF THE CHAOS

By Cédric / QueST

The " chaos " is the property that characterizes a dynamic system of which the behavior in the space of phases depends on extremely appreciable initial condition manner.

A wall of certainties collapsed at the end of the century: the emergent science of the chaos came to eliminate certainty newtonienne and laplacienne of an absolute determinism of the nature. Before the advent of the theory of the chaos, the main word was " order ". Everybody thought that the Nature had to include itself of regular manner.

The determinism:

Laplace thought indeed that if, at one time, someone knew the exact and complete state of the universe, he was him possible to know the past and the future at a time. This theory, called the determinism, part of the principle that everything that composes the universe is adjusted minutely, that there is not any room for the luck. This theory has been beaten in breach. Indeed, the Moon doesn't bend to laws of Newton for example. In the same way, if someone had the possibility all to know the universe to given one instant, the quantity of information would be as it would not see and would not hear anything.

The chaos supplants the determinism:

This theory found an important echo close to the public because it décloisonne disciplines, it makes the apology of the free will. This science is said " holistique " because it considers the world like a totality. One of pioneers of this science was the mathematician French Henri Poincaré (1942-1912).

In spite of all, this science didn't know his/her/its flight that in years 70, and this thanks to the advent of the computer. It is indeed thanks to this last that one could have studied some chaotic systems.

The chaos, in the mind of the scientific, don't mean " absence of order ", it is rather connected to a notion of unpredictability, of impossibility to foresee long-term. Small differences in the initial conditions can generate some of very big in phenomena finauxes. From then on, the prediction becomes impossible.

The famous " effect butterfly ":

Considering the weak strength of computers, in the beginning of years 60, he/it was impossible to replicate the atmosphere and the celestial oceans. Also, Eduard Lorenz (1917 -), American meteorologist working to the famous PUT, had succeeded to reduce the meteorology to his/her/its simpler expression while describing movements of air and water by simple equations since it is the interaction between these two elements that makes rain and good weather.

However, one day of 1961, Lorenz decides to redo a calculation weather report but while restarting to midway. To his/her/its big surprise, the two curves only superimposed themselves to the beginning to diverge then completely.

This was not a failing of the computer but well the theory of the chaos that had entered in action. Indeed, instead of to bring in the number 0.145237 as initial condition, he/it had gone in by laziness the rounded number, either 0.145. To his/her/its big surprise, a difference of less a thousandth generated some very important differences therefore on the final result.

Fractals:

We have all learned that some familiar things have a number of measurements that can be expressed by a whole number: the right line has a dimension of 1, we evolve in a space to 3 measurements.

However, since years 70, we know that he/it exists a category of objects of which the number of measurements cannot be expressed that under shape of fraction. Benoît Mandelbrot (been born in 1924), free-American mathematician who discovered them, qualified them of " fractals ". Contrary to what one could think, these objects are very present in our daily lives (flakes of snow, clouds...). All these objects have some irregular shapes no listed by geometry euclidienne. And, besides, the irregularity that characterizes them repeats itself to all ladders. To title of example, one can speak of a coast, in side of sea.

Benoît Mandelbrot discovered indeed that more the ladder of measure decreases, more the measured length for the coast grows until to become infinite, what goes in opposition to our common sense. Geometry euclidienne fails when it has business has the no regular. However, it is the irregular that characterizes a big majority of life things! A fractal object is therefore an object whose dimension is not a whole number. Also, an object whose dimension can be a whole number and that possesses a structure whose same motive repeats itself endlessly to all ladders of enlarging.

The Nature likes structures fractals. Also, objects fractalses found one fields of application in geology, in her botany, in the physiology... Structures fractals also find again in the human body. To title of example, the structure in blood vessel ramification, of the aorta to them capillaries, is nature fractal. It is the best solution that the Nature found to store the enormous blood vessel surface inside the volume limited of the human body.

Of the turbulence:

The chaos intervenes in everything that is fluid in the world that surrounds us. For example, water draws the complicated movements, irregulars and apparently disorganized. These turbulent movements are called chaotic movement by the scientific. Are also turbulent movements of air, volutes of smoke.

The chaos is omnipresent:

The chaos is at a time present in the macroscopic world and in the microscopic world. In 1984, two American researchers of PUT it, Jack Wisdom and Gérald Sussmen candle to the point a computer especially conceived to calculate the global orbits. Thanks to him, they projected themselves 845 millions of years in the future. And there, they realized that Pluto had a chaotic behavior with regard to his/her/its orbit. But other planet quid? Do they become later chaotic?

The French Jacques Laskar used an even more powerful computer and brought in a mathematical expression (long of 150000 algebraic terms!) concerning the middle behavior of planets. He/it transported himself/itself then until 200 millions of years in the future... and discovered that the solar system all whole is chaotic!

Pluto is not therefore the exception but the rule. Laskar noted for example that the separation between two trajectories of an any planet with the different initial conditions doubled all 3.5 millions of years. Thus, two imaginary planets only deferring 100 meters to the departure would find again, at the end of 100 millions of years, to about 40 millions of kilometers.

As measures of planet trajectories are not never perfectly precise, the global trajectories have an indefinite past and an uncertain future.

In the same way, at the level microscopic, we recover the theory of the chaos. She/it invaded the world of atoms in the beginning with her quantum mechanics of the XXth century. We cannot speak of a trajectory of the electron: we will ever specify at a time his/her/its position and his/her/its speed. The electron is at a time particle and wave: he/it doesn't follow only one orbit wisely around the core of atom but he/it occupies all spaces it empties the atom.

Of the chaos in the daily life:

The chaos is as present in our lives. Some insignificant facts have then radically changed the course of our life. For example, a man rose a little later because the tooting of his/her/its wakening didn't release, it missed his/her/its appointment, lost the job that was destined him and find again to make something of radically different of that that it had foreseen. Often, events, the discounted results are different of for what one waited: it is what one often calls the " mailman X ".

The chaos also invaded the flux and the ebb of life! The chaos has, indeed, an impact on the evolution of species. To study evolutions of populations, them scientists used concept malthusien whose equation is: population (news) = (factor of growth) * population (old)

While inserting in the model of factors limiting the growth, one can think that the population, after phases of growth followed of periods of decrease, be going to reach a balance quickly: either she/it remains roughly constant, or she/it fluctuates according to a regular enough period.

It is exact for a moderate growth rate of the population (understood in fact between 1 and 3): the population remains steady. But, when one passes the value 3, things hurry: the balance is broken and the population oscillates from one year to the other between two distinct values.

If the rate is even more elevated, the double oscillation becomes quadruple. In spite of this complex behavior, the same values come back periodically. The regularity disappears when the growth rate passes 3.57: the chaos takes the then over.

While changing some parameters to minor first view, one gets at the time of simulations by completely unforeseen thing computer.

The chaos also finds again in the Stock market. The dream of all speculator is to be able to predict the future. The financial markets include the self-regulating mechanisms based on a mixture of human psychology, social behavior and rational thought. The existence of these self-regulating mechanisms has some important and surprising implications on the behavior of markets, prices and savings, and can drag the chaos.

Conclusion on the theory of the chaos:

The chaos invested the economy, the biology, the ecology and the physiology therefore. But he/it is quantitatively very more present in the sciences said " hard " (as astrophysics...) than in sciences said " soft ". There are several reasons to this difference. In the hard " systems ", approximations are too simplistic. In the same way, the own of the adaptive " complex " systems is that they learn, remember and adjust. For example, in economy, the theory of the " adaptive anticipations " of Milton Friedman (Nobel prize 1976) and the theory of " rational anticipations " of Robert Lucas (Nobel prize 1995) are examples of adaptive complex systems.

The chaos gave the liberty therefore to the Nature: this one can exercise his/her/its creativeness therefore, too long unknown, underestimated too long...