Synergetics (Greek: "working together") is an interdisciplinary field of research originated by Hermann Haken in 1969 (see also Haken and Graham 1971). Synergetics deals with material or immaterial systems, composed of, in general, many individual parts (Haken 2004, see also Springer series in Synergetics, about 80 volumes). It focuses its attention on the spontaneous, i.e. self-organized emergence of new qualities which may be structures, processes or functions. The basic question dealt with by Synergetics is: are there general principles of self-organization irrespective of the nature of the individual parts of a system? In spite of the great variety of the individual parts, which may be atoms, molecules, neurons (nerve cells), up to individuals in a society, this question could be answered in the positive for large classes of systems, provided attention is focused on qualitative changes on macroscopic scales. Here "macroscopic scales" means spatial and temporal scales that are large compared to those of the elements. "Working together" may take place between parts of a system, between systems or even between scientific disciplines. Characteristic of Synergetics is the strong interplay between experiment and theory.
General principles of Synergetics
The systems under experimental or theoretical treatment are subject to control parameters which may be fixed from the outside or may be generated by part of the system considered. An example for an external control parameter is the power input into a gas laser by an electric current. An example for an internally generated control parameter is hormones in the human body or neurotransmitters in the brain.
When control parameters reach specific critical values the system may become unstable and adopt a new macroscopic state. Close to such instability points, a new set of collective variables can be identified: the order parameters. They obey, at least in general, low dimensional dynamics and characterize the system macroscopically. According to the slaving principle, the order parameters determine the behaviour of the individual parts which may still be subject to fluctuations. Their origin may be internal or external. Because the cooperation of the individual parts enables the existence of order parameters that in turn determine the behaviour of the individual parts, one speaks of circular causality. At a critical point, a single order parameter may undergo a non-equilibrium phase transition (see bifurcation) with symmetry breaking, critical slowing down and critical fluctuations.
Synergetics has a number of connections to other disciplines, such as complexity theory (where it is probably at least at present, its most coherent part), dynamic systems theory, bifurcation theory, center manifold theory, chaos theory, catastrophe theory, the theory of stochastic processes, including non-linear Langevin equations, Fokker-Planck equations, master equations. The connection with chaos theory and catastrophe theory is in particular established by the concept of order parameters and the slaving principle, according to which close to instabilities the dynamics even of complex systems is governed by few variables only.
Among the numerous applications of the mathematical methods and concepts of Synergetics are:
- Physics: formation of spatiotemporal patterns in lasers, nonlinear optics, semi-conductors, hydrodynamics, plasmas, geophysics, meteorology, astrophysics,
- Chemistry: formation of macroscopic spatiotemporal patterns, such as in the Belousov-Zhabotinsky reaction,
- Biology: models of evolution and development, evolution of biomolecules (Eigen-Schuster theory), morphogenesis (e.g. Gierer-Meinhardt model), growth of plants and animals, movement science (coordination between limbs and transitions between movement patterns), quadrupedal gait transitions
- Medicine: brain activities, heart beat, blood circulation,
- Cognitive Science: e.g. pattern recognition, motor control, switching among coordination states (e.g. Haken-Kelso-Bunz Model)
- Computers: self-organization, synergetic computers, attractor networks,
- Psychology: including psycho-physics, psycho-therapy (indirect control of human behaviour by changing control parameters by material or immaterial interventions),
- Sociology: dynamics of groups, collective formation of order parameters governing human behaviour including formation of public opinion etc.,
- Economy: e.g. Schumpeter cycle, competition between companies, synergy effects,
- Ecology: competition between species, impact of climate, development of forests, etc.,
- Philosophy: the concept of self-organization, strong vs. weak emergence,
- Epistemology: establishment of paradigms in the sense of Thomas S. Kuhn,
- Control theory: indirect control via control parameters,
- Electrical network theory: activity patterns, stability,
- Language theory: origin of meaning,
- Information theory: compression and inflation of information, change of information in self-organization processes,
- Management theory: indirect control of processes, corporate identity, "social climate" etc. as order parameters.
- Neuroscience: brain activity patterns, instability and switching associated with phase transitions in sensorimotor function,
The mathematical skeleton of Synergetics
Choice of variables
In a number of cases, such as laser physics, nonlinear quantum optics, plasma physics, the variables are the electric and magnetic field strengths and atomic quantities such as dipole moments and occupation numbers of atomic levels. In many cases, a mesoscopic approach is used in which many atoms or molecules are lumped together in a volume element that is large enough so that average methods can be used, but small enough so that spatiotemporal variations of locally different parts are suitably covered. Such local averages e.g. population densities or matter densities, local fluxes etc. may be used as variables in most fields. Also estimated quantities such as amount of pain experienced by a subject may be used as variables.
Equations of motion
The dynamics is described by evolution equations for the variables under consideration, i.e. the temporal change of the relevant variables is determined by the present state of the system. In general, the equations are stochastic, nonlinear, partial differential or integro-differential equations that contain fluctuations of the Îto or Stratonovich type. Quite often they stem either from the elimination of the coupling of the system to external reservoirs or the elimination of internal variables. Thereby also terms for couplings of the system with the outside such as fluxes into the system or energy dissipation may be taken into account.
Method of solution
A general solution of the evolution equations which has also to take into account initial and boundary conditions is, of course, impossible. However, the following technique has been very successful in the whole range of Synergetics: For given value of the control parameter or a set of control parameters we start from the assumption that the solution on or possibly close to an attractor is known. This may be a fixed point attractor, a limit cycle attractor, a torus, or a chaotic attractor.
Then the stability of the solution is checked when one or several control parameters are changed which in the conventional approach used in Synergetics is done by linear stability theory. The solutions of the linear stability problem essentially are of exponential nature according to spectral theory. The exponentially increasing or neutral solutions characterize the "unstable modes". Their amplitudes or phases become, in the fully nonlinear treatment, which takes also fluctuations into account, the order parameters. The equations of motions are then transformed to these new variables, amplitudes and phases defining order parameters, and the still stable modes. Then, taking into account the fluctuations, the damped (stable) modes are eliminated (slaving principle). The resulting equations for the order parameters are in general low dimensional and are of the Langevin equation type, however with nonlinearities. They may be converted into Fokker-Planck equations.
Haken, H. (1969) in Lectures at Stuttgart University
Haken, H., Graham, R. (1971) "Synergetik. Die Lehre vom Zusammenwirken", Umschau 6, 191.
Haken, H. (2004) Synergetics. Introduction and Advanced Topics, Springer, Berlin.
Springer Series in Synergetics, Vol. 1- (1977 - ) (about 80 volumes), Springer, Berlin
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Chris Eliasmith (2007) Attractor network. Scholarpedia, 2(10):1380.
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- Anatol M. Zhabotinsky (2007) Belousov-Zhabotinsky reaction. Scholarpedia, 2(9):1435.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
- Jack Carr (2006) Center manifold. Scholarpedia, 1(12):1826.
- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Hans Meinhardt (2006) Gierer-Meinhardt model. Scholarpedia, 1(12):1418.
- Mark Aronoff (2007) Language. Scholarpedia, 2(5):3175.
- James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
- 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.
Averaging, Bifurcation, Catastrophe Theory, Center Manifold, Morphogenesis, Normal Forms, Normal Hyperbolicity, Self-Organization, Self-Organization of Brain Function, Singularity Theory, Stability, Unfolding