Self-organization

From Scholarpedia
Hermann Haken (2008), Scholarpedia, 3(8):1401. doi:10.4249/scholarpedia.1401 revision #137295 [link to/cite this article]
Jump to: navigation, search
Figure 1: Snow Crystal. In the beginning of quantum mechanics and statistical physics it was believed that a crystalline structure can be calculated by determining the minimum of the free energy. This may be true, e.g. for ionic crystals, such as sodium chloride, or metals. In this case, the Schrödinger equation for the ground state or possibly low lying states must be solved. In general, this requires the solution of a many particle problem. As the example of snow crystals shows, this picture is too narrow. It is not only necessary to calculate binding forces, but rather the whole kinetics, e.g. of dendritic growth. Besides kinetics, also symmetry, may play a decisive role, e.g. the hexagonal symmetry of the snowflake is caused by the symmetry of H2O which acts as a nucleation center. This example shows that in the formation of crystals, such as the snowflake, kinetic processes and the problem of binding forces are strongly interwoven with each other.
Figure 2: A satellite photograph taken by NASA. On the left hand side cloud streets can be seen, whereas on the right hand side a vortex is formed. Cloud streets are dynamic patterns in that in the individual streets the water vapor molecules are moving upwards or downwards, alternatively. The basic question is: how do the tiny water molecules know how to arrange their concerted movements over many kilometres?

Self-organization is the spontaneous often seemingly purposeful formation of spatial, temporal, spatiotemporal structures or functions in systems composed of few or many components. In physics, chemistry and biology self-organization occurs in open systems driven away from thermal equilibrium. The process of self-organization can be found in many other fields also, such as economy, sociology, medicine, technology.

Many objects in our surrounding and daily life such as furniture, houses, cars, TV-sets, computers are man made. On the other hand, especially in the animate world, objects grow, acquire their form, and function without being created by humans. The animal kingdom abounds of examples. It is increasingly recognized that even the human brain may be considered as a self-organizing system as well as quite a number of manifestations of human activity, such as in economy and sociology. But processes of self-organization can be found also in the inanimate world: formation of cloud streets, planetary systems, galaxies etc. A fundamental question is: Are there general principles for self-organization? In the inanimate world a positive answer could be found for large classes of phenomena. In the animate world so far at least some insights could be gained. In biology (and perhaps other fields) there is a controversy: are there general principles or do we need special rules and mechanisms in each individual case?

Contents

History

The concept of self-organization was discussed in ancient Greek philosophy (see F. Paslack 1991). In more modern times, self-organization was discussed by the German philosopher Immanuel Kant (Paslack 1991), who in particular dealt with the formation of the planetary system, as well as by the German philosopher Schelling (Paslack 1991), whose discussion remains rather vague, however. In more modern times, self-organization was discussed by W. Ross Ashby (1947) and by Heinz von Förster (1992) within his “Cybernetics of second order”. It was also discussed in thermodynamics (Nicolis and Prigogine, 1977; Heylighen, 2003). A systematic study of self-organization phenomena is performed in the interdisciplinary field of synergetics (Haken 2003) that is concerned with a profound mathematical basis of self-organization as well as with experimental studies of these phenomena. The presently developing field “Complexity” (Bar-Yam, 1997) is, at least partly, also concerned with self-organisation.

Figure 3: The coordinated movements of liquids and gases leading to patterns can also be observed in the laboratory This figure shows a hexagonal pattern of liquid helium in a vessel that is heated from below. This classical experiment was first done by Bénard (1900) with oil. In the middle of each cell, the liquid rises, cools down at the upper surface and then sinks down at its border (after Bodenschatz et al., (2000)).

The laser as paradigm for self-organization of the first kind: reduction of the degrees of freedom

The first in-depth theoretical treatment of a self-organizing system concerned the light-source laser (“laser” is an acronym for “light amplification by stimulated emission of radiation”. As we shall see below, this term misses the phenomenon of self-organization, however). This example shows how man-made devices utilize self-organization processes. A typical example of the laser device is provided by a glass tube filled with molecules or atoms which represent the subsystems. At it end faces the glass tube carries mirrors, one of them semi-transparent. Due to the mirrors, only specific light waves can stay for a longer time in space between them (“cavity”) and can interact intensely with the light-emitting atoms (molecules). The atoms (molecules) are energetically continuous excited by an electric current that serves as control parameter. Below a critical value of the electric current, the device acts as a lamp. The atoms (molecules) emit randomly incoherent light wave tracks (which are amplified by stimulated emission). The light field amplitudes are Gaussian distributed. Above a critical value of the electric currents (the laser threshold) the properties of light change qualitatively and dramatically.

Laser light consists of a single wave with a stable amplitude on which small amplitude fluctuations and a slow phase diffusion are superimposed. The transition from the irregular light from a lamp to the highly ordered light from a laser shows the emergence of a new quality (namely coherent light). The emergent laser light wave is called the order parameter. Once established, it governs the emission acts of the individual atoms (molecules) termed enslavement. Since the (joint) action of the subsystems (atoms, molecules) generates the order parameter (the laser light wave), while the latter enslaves the individual parts (atoms, molecules), we may speak of circular causality.

We add for the expert that in the transition region the typical phenomena of a phase transition occur (symmetry breaking, critical slowing down, critical fluctuations that were predicted and measured in great detail). In the laser phase transition a new temporal structure is formed by self-organization, i.e. by the action of the subsystem without specific interference from the outside (no wave was prescribed from the outside, only a DC current is increased). When this control parameter is further increased ultra short pulses may occur. Generally, an instability hierarchy may occur. This is, at least in general, characterized by the occurrence of an increasing number of order parameters. Under specific conditions (including high losses due to the mirrors) laser light may show deterministic chaos (see also below). The laser is an open system with an energy input (the electric current) and energy output which besides the wanted laser light comprises incoherent losses (heat).

The just described behaviour of the laser device is typical for large classes of self-organizing systems, in which the behaviour of many parts (degrees of freedom) is governed by few order parameters (degrees of freedom).

The virtue of the laser example rests upon the fact that the laser properties can be derived from first principles: quantum theory, quantum electrodynamics, coupling to heatbaths.

Figure 4: Numerical simulation of an experiment actually performed by Bodenschatz et al. (2000). A fluid in a circular vessel, uniformly heated from below, shows hexagonal structures. When, however, the border is also heated, the liquid arranges its structure towards the formation of spirals. These spirals may be of several kinds, having one or several branches (after Fantz et al. (1993)).


Figure 5: Example of an instability hierarchy in fluids and gases. a) Liquid flow around a cylinder at small speed (Reynolds number \(r = 10^{-2}\) ). b) At an increased, critical Reynolds number, a pair of vortices is formed. c) At still higher Reynolds number, the vortices start an oscillatory movement. d) What may be called “weak turbulence” is emerging. e) Fully developed turbulence (after Feynman et al. (1965)).
Figure 6: Instability hierarchy in a fluid. In the Taylor experiment, a liquid layer is situated between two coaxial cylinders, the inner one rotating. a) At small rotation speed, a roll pattern appears, where each roll moves outwards and then inwards. b) At an increased critical rotation speed of the inner cylinder, an oscillatory motion of the rolls sets in. c) At a still higher rotation speed, the oscillatory motion becomes more complicated. d) The fluid finally shows weak turbulence, that may be related to chaos. (After Fenstermacher et al. (1979)).

Self-Organization of the second kind: no reduction to few degrees of freedom

A typical example of this kind of self-organization was given in a talk of the famous Cyberneticist Heinz von Foerster, given many years ago at a conference on marketing in Vienna. During World War II in a sea-battle between Japan and the USA, the American admiral’s ship was heavily damaged so that he could no more give orders to his fleet. Then by “self-organization” each American ship chose its own opponent. This problem is closely related to the “assignment problem” in which N machines (of partly different capabilities) are assigned to N jobs. While this problem can be solved analytically, still more complicated assignment tasks, e.g. between machines, jobs and workers are supposed to be NP complete.

An experimental access to an approximate solution may be possible by multi-agents systems (Kernbach 2008).

Self-Organization in general

Some systems may show self-organization of the first or second kind depending on boundary conditions, initial conditions, parameter values etc.

Where does it occur: Inanimate World

Physics

  • Formation of spatial, temporal or spatiotemporal patterns. We mention a few examples.
  • Lasers: coherent light, self-organization of many atoms
  • Nonlinear optics: coherent light, self-focusing, generation of harmonics, coherent Raman and Brillouin scattering, etc.
  • Fluid dynamics, gas dynamics: cloud streets, convection instability, Taylor-Couette flow, roll patterns, hexagonal patterns (Bénard cells), weak turbulences, defects, etc.
  • Gas discharges: patterns of molecular densities under the impact of electromagnetic fields.
  • Plasma physics: density and velocity patterns of partly or fully ionized atoms and electrons in (partly self-organized) electromagnetic fields, instabilities.
  • Semi conductors: patterns of electron and hole densities and currents, Gunn-effect, current filaments.
  • Astrophysics: formation and structure of planets, stars, galaxies, big bang, voids, etc.
  • Meteorology: climatology, cloud formations, cyclones, etc.
  • Geophysics / Geodynamics: inner and surface structure of the earth, geodynamo
  • Hadron plasmas: formation of hadron plasmas in high energy collisions of hadrons.
  • Self-sustained oscillations: can be found in many of the above mentioned fields.
  • Radio-engineering and other sources of coherent electromagnet fields: magnetron, clystrons, etc.
Figure 7: Pattern formation in a chemical reaction (Belousov-Zhabotinsky reaction). Out of a homogeneous stirred mixture of chemicals, spontaneously waves are formed, which run outwards and, when hitting each other, annihilate each other (courtesy A. Winfree).
Figure 8: Spiral formation in non-equilibrium systems is ubiquitous. a) Gas discharge caused by ac. b) Electroluminescence pattern in an ac-driven ZnS: Mg semiconductor layer system. c) Chemical (Belousov-Zhabotinsky) reaction. d) Optical system with Na-vapor as nonlinear medium. e) c-ADP density waves of an amoeba population. f) Ca-waves on frog eggs. (After Purwins, H.G. et al. (2007)).

Chemistry

  • Chemical reactions leading to dissipative structures (spatial, temporal structures)
  • Briggs-Rauscher reaction
  • Belousov-Zkabotinsky reaction

Robotics

  • Design of autonomous robots capable of pattern recognition, autonomous movements, language production, etc., learning. Multi-agents and multi-robotic systems.

Computers, Computer Networks

  • Internet
  • Unsupervised learning, pattern recognition (optical, acoustic, tactile) based on synergetic principles, self-programming, self-repair. The problems quoted are presently attacked by a number of groups.
  • Self-organizing (Kohonen) maps (neural networks).

Technology

  • Crystal growth, growth of microstructures and ordered defects in crystals, nano-physics.
  • Ad hoc networks
  • Traffic and material flow networks

Animate World

Biology

  • Morphogenesis, patterns on animal furs (“jaguar”), on butterfly wings, skin of fish, back bone of vertebrates, formation of body and wings of insects.
  • Growth of plants and animals, branches of trees, leaves.
  • Function of organs, e.g. kidneys.
  • Growth of brains, neural tube, neural growth factors, growth of neuronal dendrites, axons, Self-organizing feature maps (Kohonen).
  • Brain function: movement production, pattern recognition, cognition, rhythms production, speech production, generation of moods, facial expressions.
  • Population dynamics, growth, competition, extinction. Collective behavior of swarms (birds, fish) and insect colonies.
Figure 9: Spiral formation in the slime mold. The aggregation of slime mold is guided by concentration waves of cADP. The amoeba can sense the gradient and move by means of pseudo pods (after Gerisch, G. et al. (1974)).
Figure 10: The formation of a great variety of patterns on sea shells can be explained by means of reaction diffusion equations, leading first to dynamic (dissipative) structures, and subsequent solidification (after Meinhardt, H. (1990)).

Ecology

  • Survival, coexistence and extinction of species influenced by environment and mutual interaction.

Evolution

  • Mutation and selection, prebiotic evolution, Eigen-Schuster hypercycle.
  • Self-organization as a mechanism of evolution (Kauffman, 1993)

Epidemiology

  • Mechanisms of spread of diseases.

Medicine

  • Drugs as control parameters

Psychology, Psychiatry

  • How to influence the self-organization of behavior etc., indirect control.
Figure 11: The famous zoologist and philosopher Ernst Haeckel (1834 – 1919) studied among others numerous radiolarians, which are of the size of millimetres and which can form regular but also bizarre skeletons, which were drawn by Haeckel. In an individual cell, first dynamic structures are formed, which actually can be determined by methods of Synergetics. These structures then eventually solidify by means of calcification (after Blüchel et al. (2006)).
Figure 12: The sunflower head is composed of two counter rotating spirals, which must hit under a quite specific angle. The reason for this is not yet fully understood.

Other fields

Economy

  • Growth, competition, extinction of companies
  • National economy: How to steer: how to use principles of self-organization, financial markets, stock market.

Politics

  • Revolutions, “self-dynamics”, formation of public opinion.
  • Government using principles of self-organization.

Religion

  • Development of beliefs

Philosophy

  • Theories of paradigm changes (Thomas S. Kuhn)

Science

  • Development of science as self-organizing process

Theoretical Treatments, microscopic, macroscopic phenomenological

The theoretical treatment of self-organization is based both on microscopic, as well as macroscopic phenomenological approaches. Of particular interest is the question, whether there are general principles of self-organization, irrespective of the nature of the individual parts of the system. In synergetics (Haken (2004)) such principles could be found, at least for self-organization of the first kind, as outlined above. They are based on general concepts, such as order parameters and the slaving principle (for details see article on synergetics). The main issue is the reduction of complexity. In large classes of systems their dynamics can be described by few order parameters. This serves also as a basis for the application of catastrophe theory (R. Thom (1975) with a fully deterministic approach; for stochastic models see L. Cobb (1978)), as well as of chaos theory because both theories are based on the use of few variables.

The mathematical theory of synergetics provides an algorithm by which the order parameters and their (low dimensional) equations can be derived, provided the basic microscopic equations are known, and it allows one to formulate model equations in terms of order parameters if the basic equations are unknown.

While the principles unearthed by synergetics (which comes close to the spirit of dynamic systems theory) are of a rather abstract nature, another line of thought stresses the interpretation of the mechanisms (or principles) underlying self-organization. They can be traced back to cybernetics with its concepts of positive and negative feedback. According to Erdi (see also Erdi (2008)) behind many (if not all) self-organizing phenomena there is a balance between positive and negative feedback, or as H. Meinhardt (2008) (see also Gierer and Meinhardt (1972)) puts it, “self-enhancement balanced by an antagonistic reaction”. A few examples, suggested by these authors, may suffice here:

Examples 1

(based on Érdi, P: Complexity Explained, Springer 2007)

In chemical kinetics self-organized patterns occur due to interaction of autocatalytic (i.e. positive feedback) compensated by reaction step blocking the unbounded growth. The well-known BZ reactions are the paradigm. Turing structures are spatial structures, spiral waves (Winfree) are spatiotemporal patterns.

Somitogenesis, a segmentation process, which produces a periodic pattern along the head-tail axis are somewhat analogues to Turing structures, while spiral waves occur also in cardiac muscles.

Population dynamics, ecological systems: connectivity, stability, diversity, resilience, resistance. The fundamental question is how does the stability of an ecological system change if there is a change in the connectivity of the network of interacting populations, and/or in the strength of the interactions. Model studies show that weak connections enhance stability, and they ...may be the glue that binds natural communities together. (McCann et al. (1978)).

Epidemics. Epidemics is characterized by the rapid increase of the size of infected population due to consequence of the interaction between infected and susceptible individuals. Infected can converted to a removed pool spontaneously or by external control.

War dynamics. R.W. Lanchester derived and analyzed a model of classical warfare. An famous extended version of the model given by GF Gause (a biologist in Moscow) in 1934, as a model of the struggle for existence. The model is able to show the principle of competitive exclusion (one side wins, and the other becomes extinct), and coexistence, which can be interpreted as permanent war.

Segregation dynamics. Thomas Schelling’s celebrated model demonstrates how local rules (micromotives in Schelling terminology) imply globally ordered social structures (macrobehavior). Simulations results suggested that slight preference to live among their owns imply global segregation, and the formation of ghettos.

Opinion dynamics. Interaction of people in a group imply changes in their opinions about different issues may lead to consensus, fragmentation and polarization. (Remark by H. Haken: The mathematical modelling was pioneered by W. Weidlich.)

Business cycles. In a very influential model Nicolas Kaldor gave a mechanism for the generation of temporal oscillatory dynamics in income and capital by assuming a nonlinear dependence of investment and saving on income.

Self-organization in the nervous system. According to embryological, anatomical and physiological studies the wiring of neural networks is the result of the interplay of purely deterministic (genetically regulated) and random (or highly complex) mechanisms. Fluctuations may operate as ``organizing forces in accordance with the theory of noise-induced transitions or stochastic resonance. Self-organizing developmental mechanisms (considered as pattern formation by learning) are responsible for the formation and plastic behavior of ordered neural structures. Evolvability, the basis of self-organization poses constraints on brain dynamics. Stable internal representation of the external world indicate the presence of attractors. Here, an attractor means one of the states of the system where the system settles after starting from a given initial condition. Self-organization needs these attractors to have a sufficient instability to be able to alter in order to adapt to the environment. See also Self-organization of brain function.

The stable dynamic operation of the brain is based on the balance of excitatory and inhibitory interactions. The impairment of the inhibitory synaptic transmission implies the onset of epileptic seizures. Epileptic activity occurs in a population of neurons when the membrane potentials of the neurons are "abnormally" synchronized. If inhibition falls below a critical level, the degree of synchrony exceeds the threshold of normal patterns, and the system dynamics switches to epileptic pattern.

(End of Erdi's contribution)

Examples 2

(contribution by an anonymous reviewer)

Sand dunes arise from a sand deposition behind a small wind shelter; this increases the wind shelter and thereby accelerates further sand deposition - a clear self-enhancing reaction. However, sand deposited at one position cannot collect somewhere else. The depletion of the sand in the air is the long-ranging antagonistic reaction. Thus, the homogeneous sand distribution is an instable situation.

Erosion proceeds faster at some injury. More water collects in the incipient valleys, accelerating the erosion there. Meandering, also a patterning process, makes the incipient valley wider. Like in the sand dune example, water collected in a river does no longer contribute to the erosion elsewhere. Next to a large river no second river will emerge.

Localization of gas discharges as observed by Purwin and his group and mentioned by Haken are also based on this mechanism. A gas discharge leads to more ions. Under the influence of the voltage-difference these ions are accelerated, producing more ions, which lead to the further increase of the local current, etc. The resistance in the current supply and global voltage breakdown restrict this self-enhancing process. The formation and localization of a lightning has the same base.

Whether the resulting patterns are stable or oscillating depends on the relation of the time constants. If the antagonistic reaction has a longer time constant than the self-enhancing reaction, oscillations or burst-like activations will occur. An example is the time course of an infection: The infection with few viruses could be sufficient to trigger a sickness since the viruses replicate themselves (self-enhancement). The antagonistic reaction, mediated by the immune system, is much slower. It takes a day to become sick, but a week to become healthy again. (What appears on a first inspection as a disadvantage is in fact a good strategy. If the immune system would be much faster, an equilibrium between virus production and virus removal would be established. After a single infection we would fight for the rest of our life against the virus. However, due to the burst mode, we become sick for a short while but the virus is subsequently completely eliminated).

This example is also convenient to illustrate the requirement for wave formation: the self-enhancement has to spread moderately while the antagonistic reaction must be local (i.e., the condition is completely contrary to the condition for making stable patterns in space). This is clearly satisfied in the case of infections: the virus can spread, while the immune reaction is local. Therefore, infections can occur in epidemic waves. The same is true for a forest fire. Burning is a self-enhancing process since more heat releases more burnable gases, which lead to higher temperatures, triggering the burning of adjacent trees. etc. The antagonistic reaction results because the fire is no longer sustained by the ashes, i.e., the antagonistic effect is local.

(End of contribution of anonymous reviewer).

One note of caution concerning all these interpretations may be in order. In “reality”, the mechanisms can be considerably more complex, as is witnessed for instance by Meinhardt (1990), who invokes a network of different morphogenes.

Figure 13: The stripes on zebras may serve as an example for pattern formation on furs or skins of fish. The original idea for an explanation can be traced back to A. Turing (1952). He suggested that in an array of originally undifferentiated cells, bio-molecules, called morphogenes, are produced, which may diffuse and interact with each other. In this way, a pre-pattern is formed. At high concentrations of (what is now called) activator molecules, genes are switched on, which then lead to a cell differentiation, giving rise e.g. to pigmentation (see e.g. Meinhardt, H. (1982), Murray J.B. (1989)).
Figure 14: Trot of a horse The various types of quadruped gaits may serve as an example of self-organization of movements in animals. Such gaits represent specific discrete movement patterns, each of which is governed by one or several order parameters. For a detailed modelling, see Schöner et al. (1990), see also H. Haken “Principles of Brain Functioning” (1996). Upon change of a control parameter, a switch between gaits may occur.

Applications

Self-organizing systems are adaptive and robust. They can reconfigure themselves to changing demands and thus keep on functioning in spite of perturbations. Because of this, self-organization has been used as a paradigm to design adaptive and robust artificial systems (Gershenson, 2007). The main idea is to engineer elements of a system so that they find a solution or perform a desired function. This approach is useful in non-stationary or very large problem domains, where the solution is not fixed or is unknown. Thus, the engineer does not need to reach a solution, as this is sought for constantly by the self-organizing elements.

Outlook

In the science community there is an increasing awareness of the importance of the concept of self-organization and quite a number of phenomena are now seen under this aspect. Of particular interest is the development of new devices of all kinds (e.g. robots, computers) that utilize the principles of self-organization. But many further applications, e.g. crystal growth, indirect steering of population processes etc. may be expected.

References

The field is enormous and giving an approximately complete list of references is practically impossible. Therefore the readers are referred to the individual articles dealing with the different fields. Besides the historical references (Paslak 1991; von Forerster 1992) I just mention (Haken 2004) as well as the ca. 90 volumes of the Springer Series in Synergetics and monographs such as Mainzer (2002), Strogatz (1994).

  • Ashby, W. R. (1947). Principles of the self-organizing dynamic system. Journal of General Psychology 37: 125–128.
  • Bar-Yam, Y. (1997). Dynamics of Complex Systems, Studies in Nonlinearity, Westview Press.
  • Cobb, L. (1978), Stochastic catastrophe models and multimodal distributions, Behavioral Sci. 23, 360-373.
  • Erdi, P. (2007), Complexity Explained, Springer, N.Y.
  • Gershenson C. (2007). Design and Control of Self-organizing Systems, CopIt ArXives, Mexico. TS0002EN
  • Gierer, A., Meinhardt, H. (1972), A Theory of biological pattern formation, Kybernetik 12, 30-39.
  • Haken, H. (2004) Synergetics, Introduction and Advanced Topics, Springer, Berlin
  • Haken, H.: series ed.: Springer Series in Synergetics, Springer, 1977 –
  • Heylighen, F. (2003). The science of self-organization and adaptivity. In The Encyclopedia of Life Support Systems, L. D. Kiel, (Ed.). EOLSS Publishers, Oxford.
  • Kauffman, S . A. (1993). The Origins of Order. Oxford University Press.
  • Kernbach, S (2008) Structural Self-organization in Multi-Agents and Multi-Robotic Systems, Logos Verlag, Berlin
  • Mainzer, K. (2002) Thinking in Complexity, Springer, Berlin
  • Mc Cann, K., Hastings, A., Huxel, G.R. (1998), Weak trophic interactions and the balance of nature, Nature 395, 794-797.
  • Meinhardt, H. (2008) Models of biological pattern formation: from elementary steps to the organization of embryonic axes, Curr. Top. Dev. Biol. 81, 1-63.
  • Nicolis, G. and I. Prigogine (1977). Self-organization in nonequilibrium systems, Wiley, New York.
  • Paslack, F. (1991) Urgeschichte der Selbstorganisation, Vieweg, Braunschweig
  • Strogatz, S. (1994) Nonlinear Dynamics and Chaos, Addison Wesley Publ.
  • Thom, R. (1975), Structural Stability and Morphogenesis, W.A. Benjamin, Reading.
  • von Foerster, Heinz: (1992) Cybernetics. In: The Encyclopedia of Artificial Intelligence, 2nd edition, S.C. Skapiro (ed.), John Wiley and Sons, New York, 309-312

Internal references

  • Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
  • Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
  • Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
  • Teuvo Kohonen and Timo Honkela (2007) Kohonen network. Scholarpedia, 2(1):1568.
  • Mark Aronoff (2007) Language. Scholarpedia, 2(5):3175.
  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • Catherine Rouvas-Nicolis and Gregoire Nicolis (2007) Stochastic resonance. Scholarpedia, 2(11):1474.
  • Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
  • Hermann Haken (2007) Synergetics. Scholarpedia, 2(1):1400.


References to figures

  • Bénard, H. (1900), Rev. Gén. Sci. Pures, Appl. 11, 1261, 1309.
  • Bodenschatz, E., Pesch, W., Ahlers, G. (2000), Annu. Rev. Fluid Mech. 32, 709.
  • Fenstermacher, R.P., Swinney, H.L., Gollub, J.P. (1979), J. Fluid. Mech. 94, 103.
  • Feynman, R.P., Leighton, R.B., Sands, H. (1965), The Feynman lectures of Phys. Vol. II, Addison-Wesley.
  • Gerisch, G., Hess, B. (1974), Proc. Nat. Acad. Sci. (Wash) 71, 2118.
  • Meinhardt, H. (1990), The Beauty of Sea Shells, Springer, Berlin.
  • Haeckel, E. (2004), (original ed. 1904), Kunstformen der Natur, Marix Verlag, Wiesbaden.
  • Turing, A.M. (1952), Phil. Trans. R., Soc. London, Ser. B 237, 37.
  • Wolpert, L. (1969), J. Theor. Biol. 25, 1.
  • Meinhardt, H. (1982), Models of Biological Pattern Formation, Academic, London.
  • Murray, J.D. (1989), Mathematical Biology, Springer, Berlin, 2nd. ed. 1993, 3rd. ed. 2002/2003).
  • Schöner, G., Yiang, W.Y., Kelso, J.A.S. (1990), J. Theor. Biol. 142, 359.
  • Haken, H. (1996), Principles of Brain Functioning, Springer, Berlin.
  • Blüchel, K.G., Malik, F. (2006), Faszination Bionik, München.
  • Purwins, H.G., Amiranashvili, S. (2007), Physik Journal 2, 21.

See Also

Complexity, Bifurcations, Dynamical Systems, Gierer-Meinhardt Model, Morphogenesis, Synergetics

Personal tools
Namespaces
Variants
Actions
Navigation
Focal areas
Activity
Toolbox