A subsystem disturbance can be regarded as a succession of different states. Variety is the measure of the different states that are possible to that subsystem behaviour. In other words, it is a counterfactual variable.
Thus, the variety dynamics function describing the behaviour (different states) exists as a meta-function or collection of functions. This is in the same way that the function(s) describing the bounds of a set, or equivalently, the criteria for entities to be a member of that set are meta-functions that describe the characteristics of the entities that are included in that set.
Variety dynamics extends that analysis in a straightforward manner such that the variety dynamics meta-function maps to the relations between the control system varieties, the disturbing system varieties and the varieties of the behaviours resulting from the interaction between the disturbing system and the control system.
Clearly in a system context viewed in terms of the variety dynamics of multiple different characteristics of systems and subsystems, the variety dynamics functions are complex over multiple dimensions (and this is likely to be a large number of dimensions).
For conventional dynamics representation, a 2 or greater dimensional system is represented using partial derivatives. This in turn can be transformed into a complex Fourier representation or complex Taylor expansion. The variety functions equivalent of physical dynamics follow the same.
In physical systems the Nyquist ‘number’ provides the criteria for stability. For more complex multidimensional control systems, the ‘Nyquist’ stability boundary is a multi-dimensional surface that can be represented by a Nyquist-like complex function.
The ability to map variety dynamic space onto the physical dynamic space means that for an m dimensional variety space mapping a system with interacting control and disturbance sub-systems there exists a Nyquist-like function defining the boundary surface of stability for such a system.
This axiom provides a mathematical foundation for the variety dynamics field.
It is part of the formal transformation of variety dynamics from a qualitative framework to a mathematical theory with:
- Formal definitions
- Stability theorems
- Computational algorithms
- Predictive capability
- Design principles