Attractors

Consideration of the equations of dynamics of systems found that the behavior of real systems occurs in an orderly, regular manner. It turned out that nature does not allow any behavior, but only one that belongs to a finite set of possible phase portraits of the system. When describing the behavior of the system in the form of a trajectory in phase space, this is expressed in the existence of a discrete and finite set of its subdomains, within one of which the trajectory of the system behavior is located. Different subdomains differ in the specific values of the factors that determine this situation. For example, the behavior of water in the phase space “temperature and pressure” is determined by four subdomains corresponding to the solid, liquid, gaseous, and plasma states of water.

This feature of nature was manifested in the study of mathematical models of system dynamics. For example, it was found that for the systems described by Equation (8.1), there are only four types of their trajectories in the phase space “coordinates and velocities”. These types are interpreted as “centers of attraction” (“attractors”) for real trajectories of the system:

• • point attractors — stable behavior in time, displayed by a particular point in the phase space;
• • cyclic attractors — rhythmic switching from one type of behavior to another and back (usually these are oscillations between two types of action, but there may be situations when these types are more);
• • toroidal attractors — the case when the trajectories of the system behavior in the phase space are chaotically different, but remain within (or even just on the surface) the torus, the volume figure, the “tube” formed when the circle moves along a closed trajectory (like a boublik, which, however, can have not only the correct form);
• • strange attractors — “bundles” of random trajectories occupying a limited part of the phase space, but, unlike toroidal attractors, having irregular and random boundaries.

The fact that such a typology of the behavior of natural systems has a system-wide character indicates the existence of four phase states of physical substances (solid, liquid, gaseous, and plasma), and the presence of four styles of managerial behavior of managers, discovered by R. Ackoff [1]: reactive, passive (inactive), proactive, and interactive management. Examples of cyclic attractors can serve not only the oscillations of the pendulum or Buridan’s donkey but also switch of the flaps of a heavy aircraft only from minimum to maximum and back (for optimum smoothness of the trajectory), as well as periodic elections of political parties in two-party democracies (one party is for freedom, capitalism, the other for equality — socialism; these two ideas are incompatible at the same time, but can be realized alternately, upon the occurrence of a preponderance of disadvantages over advantages of one another). The transition to another attractor can also serve as a change of laminar flow to turbulent with increasing water pressure in the tap or with increasing the speed of the ship.

Further consideration of mathematical models of systems dynamics revealed that the number of attractors may differ from four and may be very large, depending on the number of degrees of freedom and the ratio between numbers of positive and negative nonlinear indicators in the equation (we can see a close analogy of attractors with different archetypes, the implementation of which depends on the relationship between positive and negative feedback loops and delays in different loops).

Attractors are forms of stable states of real systems, the final states to which the evolutionary dynamics of systems is directed. Therefore, it is natural to interpret them as objective goals of systems under given conditions (the relationship between subjective and objective goals was discussed in Section 2.3 in Part I), and the process of approaching them as a process of evolutionary self-organization. The behavior of the system in the zone of instability (near the boundary separating the zones of attraction of different attractors like a watershed of neighboring rivers) accompanied by a jump-like switching of the system from one attractor to another (such an event in synergetics is called bifurcation or catastrophe) is interpreted as a revolutionary self-organization of the system.

The discovery and study of attractors led to the realization of a number of natural peculiarities (laws) of systems. Two of them will emphasize particularly.

The first is the discovery that the randomness of real events can have not only a subjective reason (ignorance or incorrect knowledge of the subject of some objective conditions of the origin of the event) but also be quite objective (i.e., not at all related to the existence of people) feature of the behavior of nonrandomly organized systems. In the study of the mechanism of chaos in the system described by the equation without random components, it turned out that chaos often occurs with repeated doubling of frequencies in the spectrum of the phase trajectory. This is due to the hierarchical self-reproduction of the structure of the system, which leads to a feature of nature called fractality (which we will briefly discuss later).

The second is the establishment of the fact of hierarchical discreteness of the set of all manifestations of nature. In the system dynamics it is a discrete (and finite!) set of attractors of the system. In system statics, it is the discreteness of the elements, structures, and archetypes. In the physical microcosm, it is the discreteness of the set of elementary particles, atoms and molecules, and the discreteness (quantization) of their energy states. In the macrocosm, the discreteness of cosmic bodies, stars, planetary systems, galaxies. In the world of living systems, it is a hierarchical discreteness of individuals, their species, families, populations, ecological communities. In social systems, types of social formations.