**3.2 System with point attractor is half-chaotic**

The study of **the systems with a point attractor** (further – 'point attractor system', system state is not changed over time *t*, attractor is extremely short, length = 1), with parameters *s,K* = 4,3 (**met4c**) and 2,4 (**met4d**), (see more in ch.3.3, ch.4.3 and description of **Figure 4**) which make random systems highly chaotic, gave clear results - such systems **are neither ordered, nor chaotic**. Both reaction variants on a small initial perturbation (ordered - a small change in the functioning and chaotic - a big change nearby of Derrida equilibrium – **Figure 6**) appear in similar proportions (**Figure 8**). **This state was named "half-chaos"**. In this state, the resultant change in the functioning (damage) can be either very small or very large (explosions to the chaos **Figure 2**), but almost no intermediate changes (**Figures 2**–**4** and **6**). This defines a **small change** in a natural way. There remains the problem of the length and condition of the evolution of the half-chaotic system.

**Obtaining a point attractor is simple**, just after the random generation of networks (nodes connections and functions) and the states, it is enough to take that for the current state of the node inputs a node function gives the current node state. For the remain states of the input - functions stay random. The point attractor system in Kauffman terms is a completely frozen system – there is only "ice" (nothing changes). The predominance of the ice is a spontaneous property of ordered systems. Obtaining small change after disturbation of half-chaotic, point attractor system, we can expect "a small lake of activity in the ice," (originally [5]: "unfrozen islands"), which is the essence of the 'liquid' area of random systems, where Kauffman sees place for life. But such a system ceases to be a point attractor system. It turns out that the vast majority (typically over 99%) of "small changes of functioning" gives also point attractor systems. Therefore, evolution may be long, however, such the model is quite extreme and unattractive.

Simulation studies and their analysis include many important details that are unfeasible to include in this article. They are described in more than 170 pages of the report [1], Only basic ones will be listed here. The particular system is calculated at *tmx* discrete time steps *t* after disturbation, and then at *t = tmx*, more adequate value for the final results (**Figures 6** and **8**) is recorded as averaged *A* over the last 50 counting steps *t*. Due to the strong influence of various factors often sporadic, **formal errors in the obtained results are not calculated, judging such a calculation as clearly inadequate and misleading**. This problem is limited to the similarity of results from the similar simulations and the visual evaluation of fluctuations. Given here a number of networks in described series of simulations concern showed results, but often experiments were repeated in a similar way, giving a much greater certainty.

#### **3.3 The evolution from the point attractor**

Next, for models b (see ch.4.3) and c of the met4 started from point attractors we checked how long can be evolution if it accumulates small disturbances caused

#### **Figure 7.**

*The variability of basic parameters during evolution. The similarity of results for these 4 methods shows the similarity of obtained half-chaos, mainly its evolutionary stability, despite the differences in the way of obtaining. In a-c only met5c and met7ea are shown. a - Stability of parameter* q *(degree of order of the system, the contents of the left peak in Figure 6) shows lack of moving towards the chaos during the evolution accepting permanent changes which give small changes in the functioning (in the range of left peak, additionally excluded global attractors less than 7, and in the* M20 *of met5-7 also smaller than the already obtained). b - The average time of five latest explosions to the chaos (see also Figure 5a,b) does not grow in spite of the above indicated conditions on attractor's length. In the chaotic networks such explosions (see Figure 4) happen almost until the not yet exploded processes exist. c - The average size of local clusters (in met8 they are not checked) and the ice. It makes sense for in-ice-modularity, so not for the met6 where a single local cluster covers the whole network (*N *= 400). In met7e network* sf *has a specific derogation. A mechanism of it has not been elucidated (see also Fig. 1, wider recognition in [1]). d - The average number of global clusters. In the met7 it also stabilizes from the* M7*. In the initial set of initiation (*J*), still without accumulation, it is sometimes even greater than the number generated in-ice-modules, which shows that few so defined clusters may arise within one constructed in-ice-module. e - The average number of local clusters.*

small changes of functioning (small damage), but it does not allow new point attractors (**met5**). We received (also in **met6-8**) that it allows to any length of maintenances of the half-chaotic state and stabilizes its parameters (**Figure 7**). It is the **evolutionary stability** of half-chaos which **was included in the half-chaos definition**. The system still has a **significant prevalence of ice** (**Figure 7c**), and there are usually some **"small lakes of activity" forming "in-ice-modules**<sup>3</sup> **"**. Among the methods used to check the presence and properties of the in-ice-modules (see also **Figures 3c,d** and **10**), the most effective was to **track periods of node states**. The set of nodes with the same period in the process ended of accumulation was treated as a **local cluster corresponding with in-ice-module**. On average, at the same time occurred about 2 local clusters (**Figure 7e**). In the evolution, sometimes after many in the meantime accumulated changes, there appeared local clusters very similar in terms of nodes composition - a **collection of such local clusters is treated as a global cluster**. Methods to identify global clusters are very complex due to the wealth of different circumstances, including merger and disintegration of global clusters during evolution. However, we can say that they are generally quite

*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*

#### **Figure 8.**

*Half-chaos – fractions of ordered events (*q*) and chaotic (*c = 1-q*). Experiments described as in Figure 6. In the range of* q, *an order resulting from the absence of output in some nodes (*k *= 0) in the network* er, sh *and* si *is isolated as yellow. All results presented here concern only the effects of length limitation of global attractors ('6' met6) or length limitation of local attractors through in-ice-modularity. For '8' local attractors are not detected, but the level of ice (Figure 7c) shows, that local clusters cannot be large. For 'd' and '*X*' (met7a) there is no evolution, the results concern the network immediately after generation of half-chaos, but for '*X*' also after acceptance of one chaotic change, which gives a typical chaos (see also Figure 7f). In the remaining methods ('5', '6', '7' and '8') result is a sum of the results of 4 stable complete* M*, as in Figure 6, (see Figure 7). Except 'd' where* s,K *= 2,4, in remain cases* s,K *= 4,3. See also Figure 4d.*

stable formations, though they often disappear (freeze) and reappear, often in the other company of remaining global clusters, often changing period. Their average number for a set of initiations presents **Figure 7d**.

It should be emphasized that the **structure of the nodes connections in the investigated networks was constant and random**, although the randomness had various formulas that define the type of the network. **In-ice-modules are also the classic modules**, however this is only one, supporting, but less important factor. **The main property of the in-ice-modules is the activity** - changes of the states of nodes forming the in-ice-module. **The ice** (the area where the nodes do not change their states) surrounds them and **isolates from the other in-ice-modules**. In-icemodules are the result of the functioning defined by the functions and states of the nodes in a given structure. Despite the selection of functions for obtaining initial point attractor state, **functions and states of nodes had truly random characteristics**.

Simulations met4, met5 and met8 start from the system with point attractor. **In the met4** (see also ch.4.3) networks *sf* and *er* were tested. Number of nodes *N* = 400 and 4000, section *tmx* = 200 and 2000 (no variant *N* = 4000, *tmx* = 2000). One set of initialization was tested - for *s* = 2 (met4d) each node is able to one initiation, for *s* = 4 there was 3 of the remaining function values. There were gained 48,000 events for each of the three variants of (*N,tmx*). The differences in the results of these variants were not significant (**Figures 4** and **6**), for further research in met5 we used *N* = 400, *tmx* = 1000.

We limited **met5** (and next met6, 7, 8) to *s,K* = 4,3, but these studies were much more complex. For a long process of evolution (accumulation of initiating permanent changes, which give small damage) we were studied many full sets of initiations, therefore the same change in function as an initiation has been repeated, but it was separated by many accumulations. Full evolution of the particular network is a collection of 20 sets (*M*) of initiations after one initial (*J* in **Figures 2**, **7**, and **9**) set. In most of these sets, **retrogressive changes were blocked**. This results in the exclusion of a large number of initiations from the measurements and leads to a significant slowdown of evolution. After several such sets, the reversal is allowed (*M1, M7, M13, M19, M20*), assuming that the change has already another circumstance. It also allows to correctly measure of various phenomena that illustrate evolution (**Figure 7**). Since the attractor is decreasing spontaneously, making it

#### **Figure 9.**

*The difference between half-chaos and chaos in met7a and b. Experiments met7a and b (without evolution) were supposed to deeper and more accurately demonstrate the distinctiveness of the achieved half-chaotic state and chaos. In comparison to studying the evolution, an elevated* N *= 800 and* tmx *= 2000 were used. Variant b, over the conditions used in variant a, is forcing small attractors in in-ice-modules, and limitations: local attractor* ≤*100 and global attractor >200 also a shift to the latest start of local attractor <500. Experiment* J *- immediately after generation in-ice-modularity (600 networks), and after* J *further experiments* X, S, T, F *(300 networks).* X *- after acceptance of one chaotic change,* S *- after changing the node states to be random,* T *- the shift of functions to other nodes,* F *- after a generation of random functions for nodes. Despite the lacking possibility of the meaningful designation of measurement errors, the reproducibility of the results and the radical behavior otherness of* J *experiment clearly shows that the obtained state strongly differs from chaos. a,b - Probability of time of explosion to chaos for met7b. This aspect is shown in the graphs of* A(t) *shown in Figures 2 and 3 where late explosions resemble the image to the chaotic and increase the uncertainty of the appropriate selection of* tmx*. a -* J *and* X *for network* sf, ss, *and* er*. For* J *the probability smoothly decreases with time increasing, for* X *appears the collapse near* t *= 22 and the transition to a much slower decline associated with the presence of chaotic explosion after the secondary initiations. None of the collapses for the* J *results from the completion of the first round of short local attractor. After this moment there is no explosion as a result of secondary initiation inside the in-ice-module, which would be happened in the new circumstances. This mechanism is an approximation since initiations are also held in the icy walls between in-ice-modules, but there damage spreads more difficult, and after penetration into in-ice-module already subjects to the indicated mechanism. There was a clear difference in the behavior of the tested types of networks -* sf *has later explosions, in this aspect it is the most similar to the chaos;* er *has the least of late explosions. b -* J, X, S, T, F *for network* sf*. Apart from the half-chaotic* J*, the remaining chaotic* X, S, T, F *practically overlap.* X *protrudes somewhat from below, and the* S *and* T *– from above. Very late explosions also occur in half-chaos, but they are rare. These are usually cases of especially large global attractors, sometimes not at all found in the range of* tmx*, furthermore, most initiations appear in the ice between in-ice-modules, where damage normally builds up slowly. c - Average* q(t) *for* fb *(network* sf *in met7b) in experiments* J, X, S, T, F*. Half-chaos in the* J *is clearly different and quickly stabilizes* q*, but* X, S, T, F *drop up to* tmx *and probably further and are a little bit different. In this measurement the difference may be within a measurement error, which is practically impossible to determine due to the multiplicity of factors, but in d at least the* S *and* T *seem to consistently differ from the* X *and* F*. Reviewing diagrams* A(t) *as in Figure 2b similarity is noted in the range of* X, S, *and* F*, but in the case of* T *there are frequent derogation of different nature, particularly for* fa*, where the result is strongly disturbed for a few special cases. d - Average* q *for all the tested types of networks (*sf, ss, er*) and models (a, b) in all the five experiments* J, X, S, T, F*. network* er *in chaotic cases hides differences due to presence* k *= 0. See also the discussion of differences in the description c above. e - Average position for the right peak of chaotic Derrida balance. Particularly large deviation for the* Jfa *and* Jfb *is shown in more detail in Figure 6 and Figure 7c.* X, S, *and* T *behave here the nature of the derogation and the statistical derogation from the randomness of functions, which suggests such a source of visible here differences and determines the magnitude of the impact of non-randomness of functions on the results.* X *and* S *retain a correlation of non-randomness of functions with node place in the structure of the network, which* T *breaks*.

*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*

#### **Figure 10.**

*Dynamical size distribution of local clusters and their stability through evolution met7eb. Distributions at end of* M20 *of the size of local clusters in the range of up to 150 (of* N *= 400) nodes collected only in indicated sets. It should be analyzed on the greatly enlarged picture in pixels – one pixel up means one event, to right – one node more in the cluster. Dynamically observed increases are significantly more uneven than this is due to randomness, it can be assessed painstakingly analyzing the size of the growth of a specific color assigned to the particular set* M*, but it does not reflect the image of a dynamic inside the set. This non-uniformity is associated with the presence of different in-ice-modules also changing during the accumulation. Such results are practically identical in met7ea for* er *and* ss*, only* sf *clusters are there typically larger. In the* er, *larger local clusters are very rare. Presented image, especially in the dynamical form in part reproduced through colors, is a strong, eyeargument for the existence and functioning of in-ice-modules. As can be seen, in-ice-modules may even be quite large.*

difficult to move away from the point attractor, it is also forbidden to reduce the global attractor to less than 7, and in the *M20* (in met5, 6, 7) to reduce the attractor.

Parameters *q* and average time of five the latest "explosions to chaos" are the most important, they demonstrate in **Figure 7a,b** lack of converging into chaos. They stabilize starting from set *M7*, despite a slightly elevated length of global attractor was forced. There were happen that the conditions for the attractor size block further evolution. Such processes were interrupted, however, in the main series (of met5 and met6) 100 networks were obtained, which reached the end of *M20*.

It turned out that the amount of a shift (in the range of 2-50) of the point of process start (place of the initiation) after each accumulation is an important factor. We assumed a shift of 50 steps. The study was much broader and deeper, their wide description can be found in [1]. Additional attempts of evolution referral more towards the boundaries of chaos gave no noticeable nearing - a condition of acceptance of a small change is enough for any long evolution - gives evolutionary stability of half-chaos.

#### **3.4 Controlled design of the system with short-attractor**

Point attractor, as extremely short, gave sought half-chaos. However, extreme is specific and in the evolution (met5) half-chaos was maintained even when attractor was not found in the range of *tmx* (**Figure 3c**). It should be checked whether the alone condition of a short attractor, but significantly greater than 1, is sufficient. For that, simulations **met6** causing in the random system a global attractor (of the whole network) = 21 was performed. From *t* = 21 for the unused input states of the

node, the function value was changed to state 20 steps backward. **We obtained the evolutionarily stable half-chaos** even with a high *q* (**Figure 8**) for the same parameters and rules of the evolution simulation as in the met5. **The primary difference is the shape of the resulting left peak** (of small changes) in the distribution of damage size - there are practically only changes of a magnitude *A* = 0, but *A* = 1 and *A* = 2 are present in negligible amounts (**Figure 6**). This means that practically there are no changes in the functioning and in spite of the acceptance of permanent changes in the functions of nodes, nothing is changed. Such a process **is not suitable for modeling of adaptive biological evolution**, only for neutral evolution. A total **lack of in-ice-modules** was found, but the classic modules are present like in met5. In half-chaos based on in-ice-modularity as in the met5, the peak of a small damage contains a significant amount of change in the range *A* = 1 to 4, and also larger changes occur markedly frequent (**Figure 6**). Inice-modularity in met5 explains achieved stability for the larger global attractors they are assembled of small local attractors (in in-ice-modules), but this solution was checked in met7.

### **3.5 Controlled design of the in-ice-modular system**

To determine the sufficiency of the in-ice-modular state to obtain stable halfchaos, we have attempted to controlled create it without booting from the point attractor (**met7**). Networks *sf, ss,* and *er*, *s,K* = 4,3 was studied. First, a network of *N* nodes and their states are randomly generated (dependently on network type). Next, analyzing of the node connections, a collection of 'in-ice-modules' was created and everyone node was assigned to an in-ice-module or separating them ice. Node created new in-ice-module when none of its link (input and output) was connected to a node belonging to an already existing in-ice-module. When it was connected to nodes belonging to only one in-ice-module, it was assigned to this inice-module. When it was connected to the nodes belonging to several in-ice-modules or if the limit of in-ice-modules (= 10) or the size of the in-ice-module (= 100 nodes for *N* = 800, 25 nodes for the study of evolution) was exhausted, the node was assigned to the ice.

Next, a trajectory was calculated by appropriately functions selecting. For the current input state, if it was not previously defined, nodes of ice get the value of the function equal to 0, but nodes belonging to in-ice-modules – random value.

A number of additional conditions and adjustments was applied, documentation [1] contain a full description, their details are not important here. **Initially, short attractor was forced in each in-ice-module** and using this assumption basic investigations were made: (**b**) – of the in-ice-modularity state (series with *N* **= 800 and** *tmx* **= 2000 without evolution** roughly corresponding to the met4) and (**eb**) **the evolution** as in the met5 and met6 (series with *N* **= 400**, **tmx = 1000**). In the end, the necessity of **this assumption** was verified and surprisingly it **occurs unnecessary**. So the two most important research without the forcing of the short attractor in in-ice-modules were repeated (called **a** and **ea** - as logically simpler).

Examination (*J*) of the in-ice-modularity with *N* **= 800** mainly relied on checking the *q* and the distributions of damage size. In the versions b, we demanded the global attractor to be greater than 200 when the local attractor could not exceed 100 - the result was in line with the tested vision which explains the admissibility of larger global attractors. **In both versions (a and b) it was verified that the statistical properties of non-randomly selected functions are not responsible for the increase of stability**, namely - how such a system behaves after: the acceptance of one large change (*X*), randomly changing of node states (*S*), moving the functions to other nodes (*T*), and the random generation of new functions (*F*).

*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*

In the experiments *X, S,T* functions retained their statistics. In all these experiments chaos yielded (like *X* in **Figures 2b** and **8**), but it systematically slightly differed from the full version of chaos *F* (**Figure 9**).

Comparing with the met5, particular for network sf, both peaks of the distribution of damage size have been a little bit changed (**Figure 6**). Also in distributions of the ice size and the local clusters size the blur arise what caused a marked decreasing of average ice and increasing average size of local clusters (**Figure 7c**). This shows getting a **slightly different state of in-ice-modularity**. Like in the met5 and met6, system parameters stabilize from the *M7* and **the small change as a condition of acceptance is sufficient to any long maintain of half-chaos in the version of such the in-ice-modularity.**

## **3.6 Growing half-chaotic networks**

Much more complicated and stronger is the **disturbation of a system through adding or removing a node (met8)** [2]. There are problems with the comparison to the undisturbed system and the interpretation of secondary initiation. These simulations **start from a small system (***N* **= 50) with a point attractor**. The network grows in 5 successive stages *M* by 100 nodes and reached *N* = 550 at the end. The overall picture was very close to met5 and met7. Also, **half-chaos** (**Figure 6** right, **Figures 7a,b** and **8**) **with evolutionary stability and stable presence of large ice share are obtained** (**Figure 7c**). It suggests similarity of mechanisms of increased stability to in-ice-modularity. In this case, the network grows by evolving under the control of a small change. The gap between the right and left peaks is not so empty here (**Figure 6**), probably because adding or removing a node is not a very small disturbation.
