Chaos research

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The chaos theory or chaos theory refers to a non clearly circumscribed branch of non-linear dynamics and the dynamic systems , which of the mathematical physics or applied mathematics is assigned.

Essentially, it deals with orders in special dynamic systems, the development of which over time appears unpredictable , although the underlying equations are deterministic . This behavior is called deterministic chaos and occurs when systems are sensitive to the initial conditions: very slightly different repetitions of an experiment can lead to very different measurement results in the long-term behavior (chaos theory does not say that identical initial conditions would lead to different results). Chaotic dynamic systems are non-linear .

As an introductory example, reference is often made to the magnetic pendulum or the double pendulum . Other examples are the butterfly effect in weather , turbulence , economic cycles, certain pattern formation processes such as erosion , the formation of traffic jams , neural networks and lasers .

Chaos research is based on the work of Henri Poincaré , Edward N. Lorenz , Benoît Mandelbrot and Mitchell Feigenbaum , among others . The phenomena presented here correspond to the minimal consensus on what is thematically part of chaos research.


Classification of the behavior of a magnetic pendulum over three magnets; each point corresponds to a starting point for the pendulum movement. The color characterizes the magnet on which the pendulum comes to a standstill. The lighter the color, the sooner it will be. The three brightest points therefore mark the positions of the magnets.

Chaos theory describes the temporal behavior of systems with deterministic chaotic dynamics. If one tries to repeat experiments identically, this is not possible in practice, since  the initial situation cannot be restored identically due to unavoidable measurement inaccuracies - and due to noise . If a system is deterministically chaotic, the system can, after a sufficiently long time, lead to significantly different final states or measurement results despite experimentally almost identical (or, as far as possible, identical) initial situations.

This is called the “sensitive dependence on the initial conditions”. Such systems can be simulated on the computer and these simulations can in principle be repeated identically or with small deviations. The sensitivity of the initial condition occurs here in the form that if one z. B. the accuracy of the start condition changes slightly, the result of the simulation is fundamentally modified. This is due to the fact that trajectories that are arbitrarily close at the beginning no longer have this property at the end of the simulation. (Mathematically: the continuity of the mapping is given for small times, but no longer in the Limes of great times.)

In the figure opposite, the initial conditions characterized by points in the plane are colored differently depending on the final state. On the one hand there are areas (here: in the outer area) that form fractal structures , although the associated initial conditions with different end states are arbitrarily close and on the other hand there are deterministic areas (here: more inside), i.e. areas in which neighboring initial conditions all have the same end state.

In contrast to the colloquial use of the term chaos, chaos theory does not deal with systems that are subject to chance (i.e. stochastic systems ), but with dynamic systems that can be mathematically described and which in principle behave deterministically. Furthermore, the chaos theory has to be distinguished from the theory of complex systems , since very simple systems can also show chaotic behavior.


A dynamic system is called chaotic if there is an -invariant set , i.e. H. for each and every is , which:

  1. has sensitive dependence on the initial conditions .
  2. is topologically transitive to : For all open sets with there exists such that .
  3. The periodic orbits of are close in .

Limits to Predictability

If the behavior is chaotic, then even the slightest changes in the initial values ​​lead to a completely different behavior after a finite period of time, which depends on the system under consideration (sensitive dependence on the initial conditions). This shows an unpredictable behavior that seems to develop irregularly over time. The behavior of the system at certain initial values ​​(or in their neighborhood) can be completely regular if it is e.g. B. is a periodic orbit.

However, even the smallest change in the initial values ​​can, after a sufficiently long time, lead to completely different behavior, which can also appear completely irregular. In order to be able to calculate the system behavior for a certain future time, the initial conditions must therefore be known and calculated with infinitely precise precision, which is practically impossible. Although such systems are also deterministic and can therefore be determined in principle, practical predictions are therefore only possible for more or less short periods of time.

This phenomenon has also become known to the public under the catchphrase butterfly effect, according to which even the weak flapping of the wings of a very distant butterfly can in the long term lead to a different course of the large-scale weather events.

Quantum Theory, Determinism and Blurring

In the following, the determinacy of quantum mechanics (and its limits by Heisenberg's uncertainty principle ) is explained on the basis of the Copenhagen interpretation . For all other interpretations of quantum mechanics , for example the De Broglie-Bohm theory , the following section is only partially correct.

While, in the sense of classical physics, the predictability of real complex systems fails due to almost never completely exact measurements of the initial conditions, the consideration of the findings of quantum physics shows that their behavior is in principle not determined. Heisenberg's uncertainty principle says that the position and momentum of an object cannot be determined at the same time with any precision; this restriction does not relate to inadequacies in the observation process (e.g. inaccurate measurement), but is of a principle nature. This fuzziness is usually negligible in macroscopic systems. However, since their effects grow at will with chaotic systems, sooner or later they take on macroscopic dimensions (see butterfly effect  ). With the device for drawing the lottery numbers with balls, this is the case after about 20 hits. The predictability of chaotic systems therefore fails at the latest because of the uncertainty relation (because it prohibits the initial conditions from being measured with any precision). This means that, in principle, real systems cannot be deterministic in the classical sense in contrast to the mathematical models that describe them.

Nonlinear Systems

Chaotic behavior can only occur in systems whose dynamics are described by non-linear equations. Such equations are mostly not analytical; H. not by specifying explicit quantities, but only numerically solvable. The cause of the exponential growth of differences in the initial conditions are often mechanisms of self-reinforcement, for example through feedback .

If there is sufficient dissipation due to friction , chaotic behavior can usually not develop. In fairground rides, for example, which tend to behave chaotically due to their design, unexpected and unreasonable acceleration peaks could occur without appropriate braking measures .

The example of a boundary layer shows that dissipative terms do not only have a stabilizing effect . With the linear stability theory it can be shown that only the influence of friction enables the growth of small disturbances. This exponential increase represents the first phase of the laminar-turbulent transition.

Discrete Systems

So far only the temporal behavior of continuous physical systems has been considered. However, chaos is also studied in models in which each state passes discretely into the next state through an iteration step, mathematically: Examples are the logistic equation or the iteration rule that leads to Julia sets . The same basic phenomena can occur here as in continuous systems.

In principle, a discrete system can always be assigned to a continuous system by considering certain successive states. One method is the so-called Poincaré mapping , with which Henri Poincaré studied the stability of planetary motion at the end of the 19th century .


A key result of chaos research is the discovery that chaotic systems show certain typical behavior patterns despite their seemingly irregular behavior that cannot be predicted in the long term. Since they are observed in completely different systems, they are of universal importance.

Strange attractors

A typical phenomenon in chaotic processes are so-called strange attractors . To understand them, one considers the dynamics of the system using so-called phase space diagrams .

Phase space diagrams

Phase space diagrams offer a clear overview of the dynamics of a system. The state of the system is represented at any point in time by a point in a space, the coordinate axes of which are given by the set of independent state variables of the system and their speeds. The dynamics can thus be interpreted as the path of this point in phase space. For example, the phase space of a pendulum is spanned by the deflection angle and the associated angular velocity , and a periodic pendulum movement corresponds to a closed curve around the origin of the coordinates. Mathematically, the entirety of all possible behaviors can be interpreted as a flow field in phase space.


In some cases systems with different initial conditions tend to behave in the same way. The associated orbits in phase space then converge to a specific orbit, which is called the attractor . In the case of a free pendulum with friction, this would be the state of rest, i.e. the origin of coordinates in the phase diagram, to which all orbits strive in a spiral. In this case it is a point-shaped attractor, a fixed point . Attractors can, however, also be curves, such as, for example, the periodic limit cycle that occurs in a pendulum with friction that is excited to oscillate by an external periodic force . This behavior is typical for dissipative systems. From a mathematical point of view, attractors can always appear when the divergence of the flow field is negative in areas of the phase space. Fixed points and limit cycles with positive divergence are called repellers .

The strange attractor

Lorenz attractor in a three-dimensional phase space based on a simple weather model. The path point circles clockwise on the left and counter-clockwise on the right. With each revolution, the band of tracks doubles and is then cut into two halves in the middle of the image during the downward movement, whereby the decision is made whether the next revolution takes place on the left or right. The chaotic character of the tracks is based on this mechanism.

Chaotic systems can now have a special form of attractors called strange attractors. Although they are in a limited area of ​​the phase space, they are infinitely long in time and not periodic. With regard to small disturbances, they show chaotic behavior. They are fractals with a complicated and apparently irregular internal geometric structure. They are embedded in a subset of the phase space that has a lower dimensionality than the phase space itself. This means that in the dynamics, despite the chaotic character, only an infinitesimal and therefore vanishing fraction of all possible states occurs. As is usual with fractals, the attractor itself has a fractal dimension , which is represented by a broken number and which is thus even smaller than the dimension of the embedding area.

The best-known example of a strange attractor is the Lorenz attractor , which Lorenz discovered while modeling the weather. Another example is the Rössler attractor that Otto E. Rössler came across when looking at a candy kneading machine.

According to the Poincaré-Bendixson theorem , strange attractors can only appear in phase spaces from three dimensions. The reason is the fact that paths in phase space, as is usual with a flow field, do not cross, which would be necessary for chaotic behavior in two dimensions. Strange attractors can only occur if at least one Lyapunov exponent is negative and at least one is positive. In a certain sense, the negative ensures convergence with respect to one dimension and thus the reduction of dimensionality, the positive one for the chaotic behavior.

Intersections through the phase space, which are pierced perpendicularly by paths, are called Poincaré mapping . In the case of strange attractors, the intersection points form Cantor sets .

Strange attractors such as the Hénon attractor are also observed in discrete chaotic systems . Analogous to attractive structures, repulsive structures can also occur that are also fractal, such as the Julia sets .

Disturbances and resonances

Systems can react very sensitively to disturbances and thus quickly turn into chaos. It was only the KAM theorem that showed that regular influences at sensitive points in phase space do not necessarily have to cause chaotic behavior. Sensitive are e.g. B. rational (integer) relationships between an undisturbed oscillation (e.g. a double pendulum) and a periodic excitation. These cause resonances (similar to orbital resonances ), which is why only irrational relationships are considered for the theorem.

From a mathematical point of view, especially in normally prevailing measurement inaccuracies, one every irrational number can be approximated by fractions ( fraction expansion ). Hence the consideration seems practically pointless. But one has to consider that a system will build up faster due to resonances the closer the frequency ratio is to a rational value. This means that the expected values ​​deviate from the measured values ​​even faster than would otherwise be the case.

Irrational relationships that are difficult to approximate through breaks are therefore particularly stable with regard to disturbances (in terms of time). In general, one speaks in this context of noble numbers , whereby a ratio called the golden ratio is the number that can be most poorly approximated by means of continued fractions and is therefore most stable against chaotic influences.

The transition into chaos

In addition to chaos, non-linear dynamic systems can also show other behaviors, such as convergence towards a state of rest or towards a periodic limit cycle. Which behavior occurs can depend on the initial conditions or on other control parameters. A graphic representation of the corresponding catchment areas for certain behaviors as a function of these parameters is often fractal . The transition area to chaotic behavior is characterized by certain properties, such as sudden qualitative changes in behavior, which are also referred to as bifurcations .

Period doubling

Bifurcation diagram of the sequence of numbers for the logistic equation . The accumulation points  x of the sequence are shown as a function of the control parameter r. On the left the sequence converges, on the right it is chaotic, in between it is periodic. Period doubling takes place at the branch points in the transition area.

When moving from periodic behavior to chaos, a phenomenon known as period doubling or the fig tree scenario can occur . The oscillation period increases gradually by a factor of two towards the chaotic area ( in the adjacent drawing). The associated parameter intervals become shorter and shorter with increasing period ( ): The ratio of the lengths of successive parameter intervals in Limes gives the Feigenbaum constant an irrational number . The chaotic area is broken up again and again in a fractal way by intervals with periodic behavior, which in turn merge into the neighboring chaos via period doubling. This behavior and the associated numerical ratio do not depend on the details of the mathematical or physical nonlinear system. They are common to many chaotic systems.


In addition to the doubling of periods, other forms of transition into chaos are also observed, such as the so-called intermittency . In the case of a parameter value in the transition area, quasi-periodic and chaotic behavior constantly alternate, with the chaotic portion constantly increasing towards chaotic parameter values.

Examples of chaotic systems

Scientific examples

Most processes in nature are based on non-linear processes. The systems that can show chaotic behavior are correspondingly diverse. Here are some important or well-known examples:

  • The weather . At the moment the reliability of the weather forecast is limited by the rough knowledge of the initial state. But even with complete information, a long-term weather forecast would ultimately fail due to the chaotic nature of the meteorological events. The stability of the weather can vary greatly. In certain weather conditions, forecasts for a week are quite possible, in others, however, hardly for 24 hours.
  • The double pendulum . Since it is easy to model and also to manufacture due to only two independent degrees of freedom, it is a popular demonstration object for surprising changes in the chaotic sequence of movements. In computer simulations and in the tests, certain classes of system behavior can be identified, such as the maximum possible number of rollovers depending on the initial energy and the friction. The oscillating Atwood machine also has two degrees of freedom, but only one body swings like a pendulum.
  • The magnetic pendulum , in which an iron ball suspended on a thread swings over several magnets.
  • Systems with bumping balls. It is important that the spheres either collide or are reflected off curved obstacles, so that disturbances grow exponentially. Examples are the device for drawing the lottery numbers, the pinball machine and billiards .
  • The three-body problem and with it our solar system or star systems made up of three or more stars such as star clusters .
  • In medicine , the development of fatal embolisms in the case of hardening of the arteries, the failure of certain brain functions during a stroke or the development of malignant tumors after mutations of suppressor genes are typical examples of chaotic behavior.
  • The heart rhythm was at times viewed as a chaotic signal. Depending on the state of health, the heart rhythm can be classified using chaos-theoretical criteria. The parameters calculated here, however, are only empirical values. Areas of application are the prediction of sudden cardiac death or, generally speaking, the diagnosis of diseases that are mediated by the autonomic nervous system. It is assumed here that the more chaotic the behavior, the more stable the system. Viewing the cardiovascular system as "chaotic" is problematic in several ways.
  • Turbulence such as in the Bénard experiment on convection .
  • The Belousov-Zhabotinsky reaction , a chemical reaction.
  • The population dynamics in predator-prey models .
  • The baker's transformation , a discreet system that looks at the location of a raisin in the cake dough as the dough is alternately rolled out and folded.

Examples from the humanities and social sciences

In addition to these scientific examples, chaos research is also used in various humanities and social sciences to describe and explain chaotic behavior. Here are some examples:

  • Stock exchange prices and economic development . Mandelbrot had already pointed out that numerous curves of economic data have non-linear properties and can be described with the help of fractals and intermittences.
  • In historical studies , chaos research is mainly used to describe and explain crises and transition states.
  • In communication science , chaos research is used in the field of news research to better explain the selection and design of messages.
  • In psychology , chaos research is used as an approach to explain, for example, speech-psychological findings on stuttering or the causes of criminal acts of affect (such as rampages ).

However, in some cases the use of chaos theory terms in the humanities and social sciences has been criticized . The accusation is that the terms and results of chaos theory are used for the respective argumentation, although the mathematical / physical definition of a chaotic system is not or only partially fulfilled. So the reputation of mathematics and physics is claimed without any contextual connection, similar to the process of name dropping in science.


At the end of the 19th century, Henri Poincaré won a prize with a solution to the question of whether the solar system was stable. Some sources state this as the hour of birth of chaos research, but it took until the middle of the 20th century until Poincaré's approach to a solution could be usefully implemented with the help of computers .

Chaotic phenomena have been known for a long time, such as the three-body problem or turbulence . For a long time these phenomena were viewed as rather less common special cases. Since an appropriate investigation without a computer seemed unpromising, and hardly anyone expected special findings, since the phenomena are based entirely on the concepts of classical physics, they received little attention. That only changed with the advent of faster computers.

In the 1960s, Edward N. Lorenz discovered the phenomena that are now known as deterministic chaos on a model for the weather with a set of three equations for fluid mechanics. When he used rounded values ​​from an earlier calculation to save time, he observed that tiny changes in the initial conditions lead to completely different results after a short time. The butterfly effect derived from this and the formulation of the concept of sensitive dependence on initial conditions have often become misinterpreted metaphors of “chaos theory”.

In 1976 Robert May simulated a fish population with a growth rate using the formula to represent limited resources with the term . He chose a very small initial population of 2% for his computational simulation and discovered that his model computation sets in with a growth rate around a chaotic behavior.

In the 1970s to 1980s Mitchell Feigenbaum discovered the phenomena of the logistic equation and the Feigenbaum constant named after him . This equation corresponds to the Mandelbrot set investigated by Benoit Mandelbrot in 1980 , since it is also based on a quadratic equation.

Around the same time, Siegfried Großmann in Marburg and Hermann Haken in Stuttgart were working on the formulation of their theories, which were soon inspired by the ideas about Mandelbrot and Feigenbaum. Großmann formulated a description of the laser with the help of non-linear dynamics, and Haken is considered the founder of synergetics and the discoverer of the so-called principle of enslavement . The Mandelbrot set, popularly known as the “apple men”, is one of the most richly shaped fractals known.

From the 1980s, working groups were set up at many universities, such as B. in Graz, Vienna or Regensburg. In Munich, the “Chaos Group of the Technical University of Munich”, headed by Alfred Hübler , carried out numerous research projects at the Chair of Physics E13 ( Edgar Lüscher ). After Lüscher's death, it organized itself in a research association and organized a series of lecture series and several annual conferences, at which an attempt was made to represent the entire spectrum of chaos research and to enable an interdisciplinary dialogue. Large research institutes such as B. the Santa Fe Institute (USA) or the Institute for Nonlinear Dynamics in Potsdam.

Current research is more concerned with an inconsistent set of phenomena and theories. Many researchers who are still dealing with the topic today would no longer describe themselves as chaos researchers.


  • Herrmann, Dietmar: Algorithms for Chaos and Fractals 1st ed., Addison-Wesley, Bonn u. a. 1994.
  • Paul Davies: Chaos Principle. The new order of the cosmos. ("Cosmic Blueprint"). Goldmann, Munich 1991, ISBN 3-442-11469-1 .
  • Bruno Eckhardt: Chaos. Fischer, Frankfurt am Main 2004, ISBN 3-596-15569-X .
  • James Gleick : Chaos, the order of the universe. Pushing the boundaries of modern physics. ("Chaos. Making a new science"). Droemer Knaur, Munich 1990, ISBN 3-426-04078-6 .
  • Günter Küppers (Ed.): Chaos and Order. Forms of self-organization in nature and society. Reclam, Ditzingen 1996, ISBN 3-15-009434-8 .
  • Wolfgang Metzler: Nonlinear Dynamics and Chaos , BG Teubner, Stuttgart, Leipzig 1998, ISBN 3-519-02391-1
  • Gregor Morfill , Herbert Scheingraber: Chaos is everywhere ... and it works. A new worldview. Ullstein, Frankfurt am Main 1993, ISBN 3-548-35343-6 .
  • Peter Smith: Explaining Chaos. Cambridge University Press, Cambridge 1994. Standard work on the philosophy of chaos theory.
  • Marco Wehr: The butterfly defect. Turbulence in Chaos Theory. Klett-Cotta, Stuttgart 2002, ISBN 3-608-94322-6 .
  • Karin S. Wozonig: Chaos theory and literary studies. Studienverlag, Innsbruck, Vienna 2008, ISBN 978-3-7065-4507-5 , online version (PDF) .
  • Heinz Georg Schuster : Deterministic Chaos. VCH, Weinheim 1994, ISBN 3-527-29089-3 .
  • Otto E. Rössler , Jürgen Parisi, Joachim Peinke and Ruedi Stoop: Encounter with Chaos. Self-Organized Hierarchical Complexity in Semiconductor Experiments. Springer, Berlin 1992, ISBN 3-540-55647-8 .

Web links

Wiktionary: chaos theory  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. Ralph Abraham, Yoshisuke Ueda: The Chaos Avant-garde: Memories of the Early Days of Chaos Theory . World Scientific, 2000, ISBN 978-981-02-4404-0 . P. 91.
  2. Tito Arecchi , Riccardo Meucci: chaos in lasers, Scholarpedia 2008
  3. with
  4. Charlotte Werndl: What are the New Implications of Chaos for Unpredictability? (2009). In: The British Journal for the Philosophy of Science. 60, 195-220.
  5. Albert Christmann: Applications of synergetics and chaos theory in economics. Karlsruhe 1990.
  6. Otto Loistl, Iro Betz: Chaostheorie. On the theory of nonlinear dynamic systems. Munich 1993.
  7. Bernd-Olaf Küppers : Chaos and History. Can world events be expressed in formulas? In: Reinhard Breuer (Hrsg.): The wing beat of the butterfly. A new worldview through chaos research. Herne 1993.
  8. Walter L. Bühl : Social change in imbalance. Cycles, fluctuations, disasters. Stuttgart 1990.
  9. ^ Stefan Frerichs: Journalism as constructive chaos. In: Martin Löffelholz , Liane Rothenberger (Hrsg.): Handbuch Journalismustheorien. Wiesbaden 2016, p. 191 ff.
  10. ^ Rainer Höger: Chaos research and its perspectives for psychology. In: Psychological Rundschau. 43rd vol., No. 4, Göttingen 1992, p. 223 ff.
  11. Thomas Fabian , Michael Stadler : A chaos theoretical approach to delinquent behavior in psychosocial stress situations. In: Gestalt Theory, An international multidisciplinary journal. 13th year, issue 2/1991, Opladen 1991, p. 98 ff.
  12. ^ Robert M. May: Simple Mathematical Models with Very Complicated Dynamics. Nature 261 (1976) 459-467.
This version was added to the list of articles worth reading on January 14, 2006 .