Basic information
The reason for the appearance of chaos is instability (sensitivity) in relation to the initial conditions and parameters: a small change in the initial condition over time leads to arbitrarily large changes in the dynamics of the system.
Dynamics that are sensitive to the slightest changes in the initial conditions of the system, from which its development and change begin, and in which these slightest deviations multiply many times over time, making it difficult to predict future states of the system, are often called chaotic.
If the system were stable and not chaotic, then with small changes in the initial conditions, the new trajectory would not differ much from the previous one, it is even possible that the new trajectory of motion would eventually coincide with the previous one. But if the system were chaotic and unstable, then at first the old and new trajectories could be close, but over time the trajectories would become completely different, that is, the system would show high sensitivity to the initial data of the problem of motion.
Since the initial state of a physical system cannot be specified absolutely precisely (for example, due to the limitations of measuring instruments), it is always necessary to consider a certain (even if very small) region of initial conditions. When moving in a limited region of space, the exponential divergence of close orbits over time leads to mixing of the initial points throughout the region. After such mixing, it is practically no longer meaningful to talk about the coordinate of a specific particle; it is more appropriate to move to a statistical description of the process, that is, to determining the probability of finding a particle at a certain point.
Examples of chaotic dynamic systems include the Smale horseshoe and the baker’s transform .
The opposite, in a sense, to dynamic chaos is dynamic equilibrium and the phenomena of homeostasis .
Synergetics (Haken)
Synergetics is an interdisciplinary science explaining the formation and self-organization of patterns and structures in open systems far from thermodynamic equilibrium. It is founded by Hermann Haken, inspired by the laser theory. Haken’s interpretation of the laser principles as self-organization of non-equilibrium systems paved the way at the end of the 1960s to the development of synergetics. One of his successful popular books is Erfolgsgeheimnisse der Natur, translated into English as The Science of Structure: Synergetics.
Self-organization requires a ‘macroscopic‘ system, consisting of many nonlinearly interacting subsystems. Depending on the external control parameters (environment, energy fluxes) self-organization takes place.
Order-parameter concep
Essential in synergetics is the order-parameter concept which was originally introduced in the Ginzburg–Landau theory in order to describe phase transitions in thermodynamics. The order parameter concept is generalized by Haken to the “enslaving-principle” saying that the dynamics of fast-relaxing (stable) modes is completely determined by the ‘slow’ dynamics of, as a rule, only a few ‘order-parameters’ (unstable modes). The order parameters can be interpreted as the amplitudes of the unstable modes determining the macroscopic pattern.
As a consequence, self-organization means an enormous reduction of degrees of freedom (entropy) of the system which macroscopically reveals an increase of ‘order‘ (pattern-formation). This far-reaching macroscopic order is independent of the details of the microscopic interactions of the subsystems. This supposedly explains the self-organization of patterns in so many different systems in physics, chemistry and biology.
Synergetics
Synergetics is an interdisciplinary field of science that explains the formation and self-organization of models and structures in open systems far from thermodynamic equilibrium.
The basic concept of synergetics is the definition of structure as a state that arises as a result of the multivariant and ambiguous behavior of such multi-element structures or multi-factor environments that do not degrade to the standard thermodynamic type averaging for closed systems, but develop due to openness, energy inflow from outside, nonlinearity of internal processes, the emergence of special regimes with exacerbation and the presence of more than one stable state. In the designated systems, neither the second law of thermodynamics nor Prigogine’s theorem on the minimum rate of entropy production are applicable, which can lead to the formation of new structures and systems, including those more complex than the original ones. In some cases, the formation of new structures has a regular, wave character, and then they are called autowave processes (by analogy with self-oscillations).
There is also an expanded interpretation of the concept of “synergetics” in which attempts are made to extend its scope of definition to any systems, including biological, ecological, social, etc. With this approach, synergetics is positioned as “global evolutionism” or “universal theory of evolution”, providing a unified basis for describing the mechanisms of the emergence of any innovations. The expanded interpretation of the applicability of synergetic methods is also subject to criticism (see also )