INTRODUCTION TO CHAOS THEORY.

REPRESENTING ATTRACTORS BY ORBITALS.

Area of Graphical Expression in Engineering

University of Zaragoza, Spain

1. PRELIMINARY.

 In the decade of 1970 went consolidating an incipient science, Chaos, whose first stammerings date from the year 1.963, when Edward Lorenz, meteorologist at the M. I. T. (Massachusetts Institute of Technology) brought to light a curious climatic model that subsequently would fascinate many physicists by its strange behavior [2].

Nevertheless, the deep roots of Chaos, they are previous to that date, as thus it tests that have been disinterred of the oversight important mathematical works, like those of Poincare, Liapounov or Julia [2], [6].

Some years later Chaos has become the concise name of a theory that constitutes a true blooming in the scientific environment. They are frequent Congresses and international publications about it. In the United States the administrators of research programs in the Army, the C. I. A. and the Department of Energy they have dedicated budgets more and more abundant for studying Chaos [1].

Determinism and randomness were two traditionally irreconcilable concepts in the History of the Philosophy, that today, for an extensive environment of phenomena, Chaos tries to fuse and to mint in a single coin.

If we do a brief  historic review of development of science from Newton to Heisenberg, we find already the precedent that two concepts, in appearance, mutually exclusive, like particle and wave, they embrace in the breast of a more general theory, as Quantum Mechanics.

Perhaps it be not, therefore, exaggerated that someone affirm that, along with Relativity and Quantum Mechanics, Chaos is the third and last great theory, of the 20th century. Others, the most daring, they do not doubt in affirming that determinism and uncertainly are only the two faces of a same coin.

Sometimes Chaos has been defined as a science of the totality, therefore, versus reductionism of pure sciences and superspeciality of applied sciences, Chaos opposes its integrative and universalistic spirit. All of an unlike variety of human knowledge fields have been transcended already by Chaos Theory; and probably many other in the future will be it. They are not freed of their influence fields as diverse as Engineering, Medicine, Biology or Economy [1].

From view point of Engineering and, in general, applied sciences, Chaos Theory should be understood as a new tool of analysis that permits technician to confront problems up to now unapproachable or hardly to analyse by Statistics.

Without doubt, in the professional life of every engineer, they will have been presented some time problems that involve questions impossible to respond.

When a material breaks, what laws govern the propagation of the fracture through the material?

When in a great business damages in the machines arise, they do not be used to presenting remote and their temporary distribution with great difficulty can be enclosed in a statistical diagram of acceptable dispersion; really they are fruit of chance or they respond to some law?

When in an assembly line of a product with a complex process of manufacturing a faulty series arises that mocks every quality control and  leaves to market, would be able to have predicted in time similar disaster?

Here are some questions that only Chaos can aspire to respond.

Chaos is a theory of "process" more than of "state", of "to occur" more than of "to be". It is a matter of studying the peculiar behavior of certain dynamical systems, well understood that this concept (dynamical system) transcends the framework of Physics in which normally is fit in.

Inside Chaos Theory, dynamical system can be:

- For an economist, a financial market.

- For a medical doctor, a human heart.

- For an engineer, a complex network of electric energy.

A fundamental concept of this theory is that of "attractor", that appears to represent the evolution of dynamical system in the so called phases space. This type of representation was known since time ago. Already the mathematician H. Poincaré utilized it to represent "fixed point" and "limit cycle", as permanent solutions of certain differential equations pertaining to certain dynamical systems [8], [11].

So much fixed point as limit cycle are attractors, but the chaotic dynamics is characterized for a third type of attractor, that F. Takens and D. Ruelle in 1.971 they baptized with the suggestive name of "strange attractor", whose pecularity is the to possess a fractal dimension, concept that clarify in the following section.

2. THE FRACTAL DIMENSION.

All we have an intuitive concept of dimension. About a point we say that does not have dimensions; about a segment, that has one dimension; a flat figure, two dimensions and a polyhedron, three. Thinking about an object that have, for example, more than two-dimensional but less than three, is something that escapes from sensitive intuition. It was this, exactly, the abstract thought of Hausdorff, whose theoretical approaches subsequently were developed by Besicovitch. Thus it was born, in a mere theoretical plane, the concept of fractal dimension. The subsequent question was if objects that to possess this dimension would be able to be given in the reality, at least in an approximate way.

Fig. 2. a The concept of fractal dimension admits multiples definitions. All of they are seemed, and constitute generalizations of clasical concept of integer dimension, which recovers. Perhaps the most intuitive and simple to understand be homothetic dimension, that is defined like the inverse of the exponent that one must elevate the cardinal N, of a homotetic partition, so that comply the equality r = N^-1/D . For the trivial examples of the figure, dimensions are 1, 2 and 3, but for other objects more complicated, homotetic dimension results to be a fractional number.

The subsequent works of various mathematicians, among them Mandelbrot, they showed that not only can be given these objects, but the geometry of the nature is eminently fractal [7].

The term "fractal" was minted by Mandelbrot, which, reluctant to mathematical strictness, defines fractals objects as ones that possess the quality of "self-similarity" or "symmetry of scales" [1]. This property signifies, that the object at issue offers the same aspect observed to different scales; that is to say: a part of him is similar to total. Perhaps the paradigm of fractal be the one that was qualified as "the most complex object of Mathematics" and that carries the name of his discoverer: the Mandelbrot’s set (figure 2b). This set is generated for computer by means of a iterative process. Carrying out consecutive "zooms", similar replicas reappear time and again, but not identically to total.

What has awoke particularly the interest of mathematicians is that Mandelbrot’s set is similar to a collection of infinite variety of Julia’s set [6].

Thus, Mandelbrot’s set encloses great paradox that the amount of necessary information to describe it is infinite, but it is not the necessary information to generate it, what fits in some few computer code lines [5], [9].

According to Mandelbrot, fractal geometry represents the transition from order to chaos.

An strange attractor is an image or graphic in the phase space of some concrete chaotic system. It comes to represent the long-term behavior of system.

The first strange attractor was the now famous Lorenz’s attractor, so called in honor of his discoverer.

In 1.963 E. Lorenz investigated the deep reason that the weather were impossible to predict long-term. He presented a mathematical model for simulating by computer the weather [3], [10]. The model at issue was based on fluid convection; Lorenz simplified it almost abusively to adapt it to his computer, but preserving its true essence: non linearity. It was how he discovered one of the most singular properties of chaotic systems, so called "sensitive dependence of initial conditions". It signifies this property that, leaving from two points in phase space, the two corresponding paths finally diverge from each other, in spite of points at issue be as next as we want. These two points represent each assemblies of initial conditions, being paths different evolutions of system, as be the starting point [1], [2].

As every physical measurement of initial conditions of any system will involve, inevitably, a degree of imprecision, in spite of this be small, impossibility of long-term prediction of system is understood, whenever this possesses sensibility to initial conditions.

In the Lorenz’s climatic model, this verified that negligible variations of entrance, comparable to a small local squall, they were converted, in brief time, into enormous output variations in his computer. This is what, in humorous tone, is called "butterfly effect", namely, the fact that if today agitates, by its wings, air of Beijing a butterfly can modify the climate of New York next month [1].

Later, important mathematical works have happened that show singular properties of strange attractors, as well as the way to study them and mensurate in practice [2], [3], [4], [8], [11].

If impracticability of long-term forecasts were the unique thing that could be inferred from chaotic dynamics, it would lack practical interest strange attractors. Nevertheless, it is not thus. Continuing our parallelism with Quantum Mechanics, sensibility to initial conditions supposes a fundamental limitation, only comparable to Heisemberg’s Uncertainty Principle (or Principle of Indeterminacy). Just as in Quantum Mechanics already no sense makes to speak about electron orbit, but orbital, also in Chaos lacks sense to speak about orbit or trajectory of dynamical system in phase space, on the contrary one have to think about an orbital, and this would be, exactly, the atractor that governs system dynamics.

Sensitive dependence of initial conditions is a fundamental limitation comparable to Principle of Uncertainty. Just as in Quantum Mechanics makes no sense to speak about electron orbit, in Chaos has little felt representing of orbits in phase space. At figure the same three previous projections of the Lorenz’s atractor can be seen, but now represented as orbitals, that is to say, law of probability distribution.

To see more examples of attractors represented as orbitals, click here: "Images from Chaos".

An strange attractor comes to be, therefore, similar to function of probability density for possible states of dynamical system. Just as an atomic orbital, an attractor is a cloud of probability, denser in those zones in which system passes with greater frequency.

From a practical point of view, as an engineer confront  some "dynamical" problem, especially intricate, should know to recognize the authentic sign of Chaos. That is to say, should know to discern among a chaotic determinist behavior from other random phenomena of stochastic nature. For that, it has got refined mathematical tool, result from researching work of Grassberger and Procaccia, that permits to identify chaotic dynamics and to extract first conclusions on topology of presumed strange attractor [2].

The following step is the elaboration of the mathematical model for simulating by computer.

Once this is done, an engineer has sophisticated mathematical and algorithmic methods to extract conclusions and to do prognostic, useful moreover, on time evolution of system [2], [3], [4], [11].

What is exposed subsequently is an illustrative example of Chaos Theory based analysis. The chosen system for this study was some what, initially, attracted more attention from investigators in Chaos: a toy so called "Space Balls".

4.1. Time graphics.

Once mathematical model is elaborated, in first approximation, is introduced in computer and theoretical results with experimental are compared. It is used to being gratifying to see, that after some readjust of initial model, the behavior predicted by the computer coincides qualitative and quantitative with that observed in experience.

Many versions of Balls of  Space exist. For the present example a variant one has elected itself conformed by a single pendulum and a rotor with mutual magnetic coupling, (figure 4.1a).

Mathematically, this system was modelled by two EDOs (Ordinary Differential Equations) of second order. Numerical integration was done by mean of an algorithm of Runge-Kutta implemented in Pascal. That algorithm was linked to another graphical program. (Figure 4.1.b).

In these time graphics, first of all, aperiodic oscillation, characteristic of Chaos, was observed.

4.2. Sensibility to initial conditions.

The sensitive dependence of initial conditions easily can be verified and  evaluated without more than superimposing two graphics that differ lightly in initial conditions.

Here, time horizon has been called  maximum interim whose in advance can be predicted behavior of system without exceeding an admissible maximum prefixed error. As reference data we say that:

For a difference in initial conditions on the order of 10-3 mts/s and 10-3 radianes, for an admissible maximum error of 10%, time horizon is 5.6 seg (figure 4.2).

When the difference in the initial conditions is diminished to 10-6, time horizon only enlarges to 20.5 seg., for the same admissible error.

The previous thing signifies that with that precision that provide the best instruments in measurement of initial conditions, system in a maximum advance of 20.5 sec. only can be predicted.

4.3. Causes of strange attractors existence.

With a simple numerical experiment, it is easy to verify a well known theoretical consequence, what already had been aimed by E. Lorenz [1], namely, that a cause of the strange attractors is presence of disipative terms. Really, if those terms are suppressed in differential equations, strange attractor disappears and in its place a limit cycle appears (figure 4.3).

4.4. Representing attractors by orbitals in high resolution.

The following examples were carried out in graphic screen of high resolution (1.280 x 1.024 pixels) and palette of 16 million of colors.

Being formed an approximate idea of attractor topology, it can be obtained, in first place, its orthographic projections on coordinate planes XY, XZ, YZ (figures 4.4.a, b, c).

Fig. 4.4.a, b, c: Projections on coordinate planes of strange attractor for system "space balls".

Here is not represented a trajectorie into phase space, but a orbital, that is to say, laws of probability distribution (proportional to color tone)

Nevertheless, due to the complex geometry of attractors, those previous projections are insufficient for their study, as they do not reveal their complex internal structure.

A method of better viewing than the previous one consists of performing, on an attractor, a Poincare’s section. This consists in making a cut at attractor by a plane (figure 4.4.d), visualizing what remains intercepted by that plane. Doing adequate "zooms", of Poincare’s section, a complex fractal structure of attractor is showed.

Fig. 4.4.d. On the left, Poincare’s section (illustrating  corresponding theoretical concept). On the right, a concrete example: Section of dynamical system "space balls" and a zoom for magnifying a small zone.

All the images, carried out initially in scale of gray, they are able post-process, coloring different zones of attractor proportionally to their density of probability.

The conclusions are varied, but they can be classified in two groups:

- Discrimination of different zones of probability of attractor, that go from zones of null probability to zones of high probability that system cross them.

- Variation of parameters of mathematical model to see as influences in attractor topology.

In short, can tell about this method of analysis, that is directed to condense in a global way, in images, all of enormous amount of information generated by computer. Suffice to say, that elaboration of one Poincare’s section, computer calculates various hundreds of millions of points. For a power of calculation of 12 MIPS, this supposes, at least 8 hours of work of CPU. If instead of presenting them by graphic screen, numerical generated results would be printed, various thousands of sheets would be filled (time graphics would save sheets, but would be a too long paper roll). It is understood that interpretation of information by a user, thus presented, would be unrealizable task.

A second phase of investigation would consist of rough estimating fractal dimension of  atractor, the Liapounov’s characteristic coefficients(LCEs) and the K-entropy (Kolmogorov’s entropy) as measure of disorder generated by the system [2], [11].

Bibliographical references.

[1] CAOS: la creación de una ciencia.

J. Gleick, Seix Barral, 1.988.

[2] El orden caótico.

M. Dubois y otros, Mundo Científico, vol. 68, 1.987.

[3] L'ordre dans le chaos.

P. Bergè y otros, Hermann, 1.984.

[4] An introduction to Chaotic Dynamical Systems.

R.L. Devaney, Addison-Wesley, 1.989.

[5] The Science of Fractal Images.

M.F. Barnsley y otros, Springer-Verlag, 1.988.

[6] El lenguaje de los fractales.

Heinz-Otto Peitgen y otros, Investigación y Ciencia, vol 169, 1.990.

[7] Los objetos fractales. Forma, azar y dimensión.

B.B. Mandelbrot, Tusquets editores, 1.984.

[8] Nonlinear Differential Equations and Dynamical Systems.

F. Verhulst, Springer-Verlag, 1.990.

[9] The beauty of fractals.

H.O. Peitgen y otros, Springer-Verlag, 1.986.

[10] Fractals in the phisical sciences.

H. Takayasu, Manchester University Press, 1.990.

[11] Chaotic Dinamics of non linear systems.

S. Neil Rasband, John Wiley & Sons, 1990.