**1. Introduction**

This is empirical work1 using simulation. It concerns dynamics in complex autonomous Kauffman networks that are finite and discrete, shows that current theory used for them, based on Lyapunov coefficient in infinite, continuous space, implies false expectations.

Kauffman [5, 6] has considered as a general model of a system an autonomous, dynamic, deterministic, complex, random Boolean networks (known as RBN). The discovery of chaos, order, and phase transition between them in such the networks, allowed to look into this very complex world. Lot of works are based on this model [5, 7–17]. Now, slowly become aware of new important aspects that we have not yet considered adequately. In this paper, such the way is developed, but considered networks are not fully random and use more signal variants than only two. A new obtained vision is clearly different and more adequate for description of adapted objects than now widely accepted. The discovery of half-chaos is the main new element here, which above all frees from strong limitations in systems modeling imposed by the contemporary vision. The statistical properties of the systems are easiest to investigate for fully random systems and from these, we should have started (like Kauffman did), but the systems we model are usually suited to some tasks and are certainly not completely random.

Lyapunov exponents are the most widely used measures to describe chaotic behavior of dynamical systems, however, to check an adequateness of theory by observation of the behavior of finite dynamical networks it must be defined using its main features expected by the theory. The main characteristic of the chaotic behavior of dynamic systems is a high sensitivity to initial conditions, leading to maximally different effects for very similar initial conditions. A small disturbance is a small change in initial conditions. An effect of such small disturbance is called damage. Distribution of damage size [6] ("size distribution of avalanches" in [17]) is then the main feature observed in experiment and expected by the theory that may be compared. It is original; theories using Lyapunov exponents or percolation are derivative. The term 'chaos' is used here in such the meaning, similarly as Kauffman does (see ch.2.3). To fit the current theory the damage distributions should fit a Derrida's annealed approximation model [18] and for chaotic systems an equilibrium level found in it.

It is commonly believed, that system can be only either chaotic or ordered, but not simultaneously both of them - this is shown to be false. On this believing, a "criticality hypothesis" (critical regions have also been said to be "at the edge of

<sup>1</sup> The description of the investigation and the arguments for introducing the half-chaos given in this article is necessarily shortened and simplified. A much more extensive description is available in supplement [1, 2] to this article. Earlier, simpler versions of the article are available in preprints [3, 4]. The data (programs and its sources, results of simulations) analyzed during the current study are available from the author on any request.

Wider list of abbreviations and new terms is placed on the end.

In this work a few abbreviations that are not standard are used: *s* - number of equally probable signal variants.

Network types: *sf* - scale-free; *ss* - single scale; *er* - Erdős-Rényi "random"; *sh* and *si* are respectively *sf* and *ss* with 30% removal of nodes.

*tmx* - maximum number of counting steps of time *t*;

*dmx* - mean maximal damage *d,* i.e. Derrida equilibrium for chaotic behavior;

*q* - degree of order, fraction of damage which are a small change of network functioning at tmx, capacity of left peak of *P(d)*.
