*Life Is Not on the Edge of Chaos but in a Half-Chaos of Not Fully Random Systems. Definition… DOI: http://dx.doi.org/10.5772/intechopen.93864*

chaos" in parameter space of systems) is formulated (see e.g. [17]). Evolution needs small changes which practically occur only in critical regions in such the systems. Current theory and this believing (see e.g. [17]) are based on the assumption that networks are fully random. However, interesting phenomena concerning life occur in not fully random networks due to natural selection. The current theory of chaos was built for functions in infinite and continuous space, but it is used for finite discrete networks [7, 8], such a method is an approximation. It loses a few important phenomena present in such the networks, but absent in the infinite and continuous space. Due to such the reasons, expectations of the theory that life is on the edge of chaos can be and are inadequate. Here such phenomena are shown; they need much more complex theory which will not use the assumption of full randomness of network and infinite continuous space, but to build such theory is the next step, which is the task for mathematicians. The description of this experiment in the language of mathematical equations seems to me inadequate and unattainable, and in my opinion useless, but mathematicians may have a different opinion. Programming languages are a natural and appropriate tool for describing such issues. I can share the program, but it is complex. It is not true that the below description of experiment is not exact enough to be repeated by every IT specialist. Therefore it is enough exact to understand by mathematicians too.

Indication of adequate ranges of parameters of a complex "purposeful" (adapted) system describing living, technological or social object is a key for modeling their processes. An important parameter is a connectivity [19], which current theory strongly limits. The system can be any, e.g. the solar system is also a system, but usually, in human intuition, the system has to somehow work (therefore above "purposeful"), and despite some changeability, it has to keep its identity. The evolution of the system is a term that reconciles two adversities - variability that is the essence of evolution, and the identity of the evolving object. This is not a philosophical problem, but a particular problem for modeling. In this work a base for solving this problem is found. A good approximation of the system description is a dynamic complex network, although it undoubtedly has many important simplifications. We are just entering this subject and it is difficult for our intuitions to operate on more complex, more adequate descriptions, such as process algebras [20].

Half-chaos is a state of the system that is not fully random, with parameters that make the random system strongly chaotic (hereinafter we will call them "chaotic parameters", such the parameters are usually estimated for real systems), however small disturbances give an ordered reaction (small damage) with a similar probability to a chaotic reaction (damage near the Derrida balance [18], **Figure 1c,d**). Acceptance of changes that trigger ordered reactions preserves the half-chaotic state allowing for a long evolution of the slowly changing system (the system retains identity), but acceptance of one change that gives a chaotic reaction leads to practically irreversible entry into normal chaos (the system works completely different, ceases to be itself). Thus, the basic Darwinian mechanism emerges - this has large interpretational consequences.

The assessment of whether a given system is chaotic or ordered is currently based on parameters that in the case of a half-chaotic system indicate chaos for the fully random system (I call them "chaotic parameters"), but the behavior of the system turns out to be inconsistent with such prediction. This work presents halfchaotic systems and simple ways to obtain such systems. The experimental results are unambiguous and easy to repeat. The constraints forming the half-chaotic system are small, which means that there are a lot of such systems, though undoubtedly significantly less than of fully random.

The practical result of this work is the realignment of the acceptable range of parameters for system modeling. This is a fundamental change. First and foremost,

#### **Figure 1.**

*Comparison of models based on* p *and* s; *- of influence of* s *and* K *on Derrida equilibrium, Derrida plot;* d(t)*. a - Comparison of models based on probability* p *of one Boolean signal variant and on* s *equally probable signal variants in dependency on* K*. As the basic argument* s *is taken. For it* p *is added as 1/*s*, it is for the case if in reality there are* s *different equally probable signal variants, but we are interested only in one of them, and rest we collect to the second one. Values of the coefficient of damage propagation* w*<sup>s</sup> for* s,K *and* wp *for* p *and* K *are used. The equation for* wp *is taken from [5, 14]. Both models give very different results, it means, that they cannot replace each other. (See also ([21] Fig. 4)). b - Derrida equilibrium (*dmx*) for chaotic response in the system of* s,K*. Kauffman using Boolean networks has considered only* K *as the most interesting variable, but* s *influences* dmx *more hardly. However, he cannot use* s *other than 2, because for each s > 2 the chaos is present (*dmx *> 0 exists), like for any* K *> 2. Among sensible* s,K*, only for 2,2 exist order, it is an especially extreme case. c,d - theoretical damage spreading calculated using the Derrida's annealed approximation model. d - The change of damage in one step of the time in synchronous calculation known as the 'Derrida plot', extended [21] for the case* s *> 2. The crossing of curves* dt+1(dt, s, K) *with diagonal* dt+1 = dt *shows equilibrium levels* dmx *up to which damage can grow. Case* s,K *= 2,2 has a damage equilibrium level in* d *= 0. These levels are reached on the left which shows damage size in time dependency. For* s *> 2 they are significantly higher than for Boolean networks. All cases with the same* K *have the same color to show the influence of* s*. c - In this plot expected* d(t) *for* N *= 2000 is shown. It is an effect of 'Derrida plot' shown in d. A simplified expectation* d(t) = d0ws <sup>t</sup> *based on coefficient* w*<sup>s</sup> is shown for the first critical period when* d *is still small - three short curves to the left of the longer curves reaching equilibrium. Parameter* **s***,***K** *(treated as a vector) is the main variables in the simulations. Most of the studies are made for* s,K *= 4,3, also sometimes for* s,K *= 2,4 (that is, for Boolean network). They provide highly chaotic random systems - 'coefficient* w *of damage propagation' is significantly higher than one.*

"chaotic parameter" for the Kauffman (Boolean) network - connectivity is included in this scope, but also a larger number than two of signal variants, also omitted due to the effect in the form of a chaotic system (for fully random systems). The need to introduce a larger number of signal variants for statistical investigations was already explained in [21]. The maintenance of the name of the 'Kauffman network' for such a network was there postulated, to be no longer synonymous with Boolean networks. However, these postulates acquire practical significance only after demonstrating half-chaos.

*Life Is Not on the Edge of Chaos but in a Half-Chaos of Not Fully Random Systems. Definition… DOI: http://dx.doi.org/10.5772/intechopen.93864*
