What are the characteristics of chaotic systems

Deterministic chaos

In deterministic chaos, the behavior of systems becomes unpredictable, although it is predetermined (= determined) by known motion sequences. Because small initial disturbances increase here over time.

At the end of the 19th century, astronomy enjoyed great success. The planets Neptune and Uranus were predicted and discovered due to discrepancies between the calculated and observed movement data of Jupiter and Saturn. Thereupon King Oscar II of Sweden set the task of proving the stability of the solar system. The award finally went to the French mathematician Henri Poincaré, but for the proof that this proof cannot be given. Poincaré discovered the so-called hyperbolic structures in the space of the places and speeds (phase space) of the planets, which make long-term tracking of the planetary orbits practically impossible: The planetary movement turns out to be chaotic on large time scales.

Pattern in stirred liquid

What appeared to be a bizarre peculiarity of planetary motion in Poincaré's time soon turned out to be an essential element of non-linear natural laws. Almost a century later this chaotic behavior was found in almost all areas of the natural sciences: Extensive experimental and numerical studies in biological, ecological, chemical, physical and other systems repeatedly produced the same irregular, seemingly random movements, the same long or even short-term Unpredictability to light. Chaos is everywhere! The future weather, for example, can in principle be calculated, but with good accuracy only for a short period of time.

What is remarkable about this type of chaos is that it does not arise from a multitude of unknown influences, such as the noise in electrical circuits caused by the disordered movements of myriads of electrons. Rather, it is a consequence of the otherwise completely deterministic (= predetermined) sequence of movements. How can this be understood? First, let's imagine a pool table with two balls some distance apart. In two different experiments, one of the spheres is pushed onto the other at a slightly different angle, but from the same position in each case. After each collision, the angle between the two trajectories of one sphere is significantly larger than before. When we think of colliding gas atoms in space, every angular perturbation in their trajectory is enlarged with every further collision with other gas atoms. Very soon the information about the original thrust was completely lost. The phenomena described are referred to as deterministic chaos, which has two characteristic, defining features: on the one hand, almost every initial disturbance of a chaotic system, no matter how small, is continuously amplified; on the other hand, a certain value range that is fixed for all system parts is not left (for example 360 degrees for the impact angle). Both together lead to irregular and unpredictable movements that look like random.

Strangely attractive

Line between order and chaos

Even in this chaos, certain spatiotemporal patterns emerge over a long period of time, which generally have a fractal structure: They are called strange attractors. Although long-term predictions of the sequence of movements are practically impossible, even with massive computer support, we have now gained deep insights into the properties of such chaotic processes and, for example, learned to understand the self-similarity and the fractal character of their attractors. The magic word here is “renormalization”, with the help of which Mitchell Feigenbaum was able to show the universality of the nature of chaos. In the meantime, this finding has been experimentally confirmed in many different situations. Here, too, universality and the laws of scale prevail. Thus a new methodological arsenal is available, which will enable the solution of many structure formation questions in nature and technology.

A characteristic sign of chaotic behavior of systems is that their passage over time can only be described by a continuum of periodic movements. The equations of motion are purely deterministic and without random elements, but they are non-linear. Chaos only requires a few degrees of freedom. But it is not limited to simple systems with a few degrees of freedom.

Chance and Statistics

Chance and statistics encounter us in nature in three ways.

  1. First, there is the statistical character of quantum mechanics, about which the strict laws for probability amplitudes give us information.
  2. Second, there is the deterministic-chaotic movement, which occurs as a result of non-linear classical laws of nature, with all known random elements such as unpredictability and irregularity in the process.
  3. Thirdly, there is coincidence, which arises from the interplay of many individual parts (electrons, molecules, grains, lumps of dust in the ring of Saturn, star clusters, etc.).

We now know that for many statistical phenomena of a macroscopic nature it is irrelevant whether the particles involved move according to quantum mechanical or classical, non-linear laws. The large number of particles involved alone causes macroscopically random behavior that we clearly describe as "noise".

As recently discovered, the noise can cause resonance vibrations and directional motion. In addition, it significantly influences the course of all chemical reactions and it could also be relevant for the functioning of the muscles.

Applications

The knowledge gained in the meantime for the analysis of chaos is also increasingly being used outside of physics, for example in medicine for the evaluation of electrocardiograms and electroencephalograms, as well as for quality controls or for the analysis of stock market prices. Deterministic chaos will possibly play an important role in future high-speed computer chips that work with "ballistic" electrons.