In what follows, we use clear linguistic formulas to express the proposed regularities as three laws, the formation, the functioning, and the decline of an autonomy.
     These three regularities is the first component of the proposed RBZ methodology. We propose a five-vector model of “space and time aberration“ (SATA) as the second component of this methodology for the description of one cyclical process, or one of the many convolutions of an autonomy. We added a fifth vector to the classic space-and-time model, the vector of aberration d.
     Let us illustrate the implementation of the SATA model using a classic pendulum as an example, noting that the existing models of the pendulum do not require a new methodology. A differential equation and several functions would be sufficient to describe the movement of the classic pendulum accounting for friction.
     We use the pendulum and the RBZ methodology to understand that the methodology is fully applicable for description of a system like an oscillating pendulum. The functioning of a pendulum has two interrelated cycles. First, the cycle of reproduction, which consists of two components. One component is a vector, which we call the vector of perspective conversion (PC). It describes the deviation of the pendulum to a certain level from the state of equilibrium. The second component is the vector of retrospective conversion (RC), which describes the transformation of the potential energy created by the perspective conversion into heat. Second, the cycle of the transformation of kinetic energy into potential energy, and potential energy into kinetic, i.e. one swing of a pendulum. Not only does the proposed vector model of the “pendulum” autonomy consist of two components, but also each cycle per se consists of two components (figure 2).
     As we move the pendulum away from zero point on the vertical axis, we create the perspective conversion (PC vector) of the cycle of reproduction. The realization of the potential energy created at the highest point is the beginning of the retrospective conversion process of the cycle of reproduction, which ends once the pendulum stops. Let us use the SATA model to understand the conversion of the main cycle in the pendulum model, that is the transformation of potential energy into kinetic and that of kinetic into potential energy. We introduce the vector of predisposition d, which consists of the time, space, and energy parameters, which change over a cycle. Upon friction with air, the components of vector d will decrease: the energy used by the pendulum over one oscillation, the volume of space (amplitude) used by the pendulum’s oscillations, and the time of each subsequent oscillation. Having two oscillations and two values of vector d, we can predict the complete model of the pendulum’s movement. The difference between the values of vector d over the first and second oscillations can serve as the forecast model of the development of the model as hand.
     It has been noted above that the vector model of an autonomy consists of the main and reproduction cycles, the latter creating the vector of evolution. The pendulum example makes it clear that each of the two cycles consists of two components. The main cycle consists of the disturbance process (DP) and the slowdown process (SP). The cycle of reproduction consists of the perspective and retrospective conversion. In this case, the vector of evolution is the process caused by pendulum’s friction in the oscillation space. A model with five vector values (DP, SP, PC, RC, d) is referred to as a vector autonomy, which we propose to analyze any complex system, including the society, a family, economics, a human, complex technical systems, etc. That is the third component of the proposed RBZ methodology (figure 3).
     Everything starts with two interrelated cycles, or with two interrelated convolutions, to be precise. The information on one convolution can be used to create a forecast model of the development of a complex system. The information on the vector of aberration over one of the parameters of time, space, or energy – over one oscillation – can be used to describe the whole system until its termination.
     The world of complex systems, which develop through space and time as one whole, becomes clearer and simpler for us if examined through the paradigm of vector autonomy. The existence of such complex systems as the society or a human organism with millions of processes suggests that each of these cycles in the process of evolution became a hierarchy in its structure. The hierarchies of the main transformation cycle and the reproduction cycle are simultaneously created in such systems. And however complex the system, vector autonomy lies in its foundation, which consists of only five vectors. With the oscillating pendulum as an example, we can trace another regularity. That is, the duration of a system’s functioning is determined by the perspective conversion, which is the energy created when deflecting the pendulum, and the vector of perspective conversion, which is created during one pendulum oscillation in the oscillation space. The perspective conversion is bounded by the horizon line, to which the pendulum can be deflected, and the retrospective conversion ends when the oscillations stop.