It is well known that nonlinear systems can display both regular and chaotic behavior depending on controlling parameters. As all complex systems are nonlinear, therefore they also have the common property of universality on the edge of chaos. In this paper the use of this universality as a paradigm for complex systems simulation is proposed.
If we have a complex system whose formula is unknown in detail, one would think it is impossible to determine with any certainty its ultimate behaviour. However, one of the main themes within the field of Chaos Theory is the universal behaviour of complex systems on the edge of chaos where the main features of the "outward" behaviour are not dependent on their hidden "inward" mechanism. Computer simulations allow us to investigate the universal behaviour of complex systems and to apply Chaos Theory to simulate systems on the boundary between their regular and chaotic behaviours. If we are able to assign to complex systems appropriate parameters of order and chaos, then we may adequately simulate the most complex phenomena on the edge of chaos.
A tradition in perceiving our world as a relation between three independent worlds has recently received some new development. Roger Penrose [5] considers three worlds and relationships between them. One of the worlds is the mental world of the person. It is objective and real. Another world is the objective physical world. The third world is Plato's world of ideas, mathematical structures, absolute truths, etc.
In the mental world there are real "things" and "phenomena". For example, there exists happiness and pain, smell and colour, love and understanding, impressions and images of stars, tables, chairs, etc. In the physical world there are the real tables and chairs, sun, stars, stones, flowers, butterflies, space and time, molecules and atoms, electrons and photons. The Platonic world also has real existence, as well as the two previous ones. Ideas and mathematical structures exist in the same sense in which things really exist in the physical world (under the physical laws) and images, feelings and imaginations exist in the mental world (maybe under psychological laws).
The crucial question is how are these three worlds connected? Firstly, physical reality submits to laws of nature that have an exact mathematical form. Physical reality would seem to be born in the body of mathematical structures. Secondly, real physical phenomena and bodies create the mental world (at least in the form of human consciousness). Thirdly, ideas, concepts, and mathematical theorems have to be discovered in the mental world to continue their "independent" existence in the Platonic world.
Plato has expressed his understanding of the connections between the worlds in his parable about the cave. From Platonic times the role of mathematics in understanding of the structure and phenomena of the physical world has been considerably increased. When Albert Einstein formulated his theories, physical phenomena which required such elaborate mathematical descriptions were unknown. Newtonian Theory provided satisfactory accuracy in the descriptions of the corresponding phenomena. Now almost nobody doubts that the General Theory of Relativity is much more precise in describing the phenomena of the physical world, as compared with the classical theory. This means that Einstein had not only improved the description of observed physical phenomena by means of mathematical structures but he had found out new mathematical structures that connected to physical reality in a specific way.
It seems to be interesting that the existential cycle may have spins - "left" (ideal, physical, mental) and "right" (ideal, mental, physical). However, there is here a more important peculiarity in that the world dynamics looks like a kind of an iterated map of an interval into itself which in some conditions displays a universal (qualitative and quantitative) behaviour on the edge of chaos. It is important that we see the period doubling effects [4], the sustainable period "three" [3], the universal metric properties in transition to chaos [2] and so on, not only in mathematical structures (the Platonic world) but also in the physical and mental worlds.
The paradigm of a universal dynamics on the edge of chaos may be formed by considering the simplest nonlinear dynamical systems (one- dimensional parametrical maps of an interval into itself) in the parameter region from which chaotic behaviour is emerging. The general phenomena of universal regularity seems always to appear near to chaos. Surprisingly, a lot of qualitative and some quantitative edge results hold in a very wide context. This means that the mere fact of adopting a certain type of simulation (for example, choosing to simulate complex dynamics by continuous maps of an interval into itself) has already made a profound influence on the outcome of the analysis.
An iterated map into itself serves as the simplest simulation of complex dynamics. This simulation presents an interesting mathematical structure going far beyond the simple equilibrium solutions one might expect. If in addition the dynamical system depends on a parameter, then there is a corresponding mathematical structure telling us a lot about interrelations between the behaviours for different parameter values.
The paradigm of universality on the edge of chaos should be seen as an attempt to isolate simplifying features of dissipative dynamical systems. When they are driven by parameters from the outside, then strange behavior quite often emerges and, particularly, a kind of universal behavior for the successive bifurcations in one parameter families of maps. Iterations of the form Xn - Xn+1 = F(Xn), where F maps into itself, can be viewed as a discrete time version of a continuous dynamical system. Here, n plays the role of the time variable. The behavior of such a system is, to some extent, independent of the particular map. For certain values of a control parameter the system moves from a regular behaviour to a chaotic one in the universal way. When the parameter changes, we can see the only possible sustainable state of the system. For a certain value of the parameter the system attractor is split from one state to two possible states. As the parameter changes further, additional splits take place at intervals dependent upon a constant (Feigenbaum number 4.669... for the one-dimensional case), giving more complex limit cycles progressively until the system enters a chaotic state. A more detailed description of the paradigm can be found in [1] and in other good articles and books on Chaos Theory.
The relations between periodic and aperiodic (chaotic) behavior which are relatively clear for the case of one-dimensional maps of an interval into itself, should help us to understand at least some parts of the more complicated multidimensional case. The choice of a one-dimensional case for the paradigm is unduly restrictive. It is true that multidimentional systems may show phenomena which simply cannot occur in one dimension due to their restricted volume of 'phase space' and/or combinatorial effects. On the other hand, some results extend straightforwardly to multi-dimensional systems. However, for methodological discussion this one-dimensional choice seems to be pertinent.
How are we to simulate the "edge dynamics" of complex systems whose "formulae" are unknown in detail? The first traditional step of the edge behaviour simulation is a mechanical junction of the regular and chaotic images and attempts to simulate these images of the edge dynamics. Such an understanding of the simulation task is analogous to the situation in nonlinear mathematics until R.May, T.Li, J.Yorke, M.Feigenbaum and their collegues described the universal properties in the region of transition to chaotic behaviour. So if the universal property which is characterized by the universal constants (M.Feigenbaum) does not appear, there will be inadequate results of the simulation, of course. Therefore, the next stage of the complex system methodology is one that accounts for both the universal qualitative property and the universal quantitative property of the edge behaviour.
There are enough grounds to assert that all the complex dynamical systems (many-dimensional and many-parametrical) also have the same universal properties in behaviour on the edge of chaos. These properties can be expressed by means of various mathematical forms but that it is unimportant here. Here we are interested in meta-theoretical and methodological questions. So, the clue is that understanding that there are universal metrical and qualitative properties in the parametrical transition between regular and chaotic attractors is the effective methodological heuristic in the searching for simulation algorithms whilst a concrete "inward" mechanism which gives rise to the complex edge behaviour is not known.
A conception for development of expert systems based on this methodology may be also proposed. If an expert system based on the paradigm of the universal behaviour on the edge of chaos is built, then it is necessary to fix the relation between basic elements of the system and key notions of the paradigm. A system developed on the paradigm would demonstrate good results in simulation to the edge of chaos whilst indifferent to the completeness of description of the concrete mechanism ("hidden inward parameters") that caused the complex behaviour, because of the clue that properties of all the complex systems on the edge of chaos seem to be universal. This is why I think we can understand each other in spite of being so complex, and so different, and so close to chaos.