New possible properties of atomic nuclei investigated by non linear
  methods: Fractal and recurrence quantification analysis
New Possible  Properties of Atomic Nuclei Investigated  by Non Linear Methods, 
Fractal  and  Recurrence  Quantification Analysis. 
             
                                                               Elio Conte(*) 
(*) Department of Pharmacology and Human Physiology – Tires – Center for Innovative Technologies for 
Signal Detection and Processing, University of Bari , Italy; 
International Center for Studies on Radioactivity, Bari, Italy. 
                                                            Andrei Yu.  Khrennikov (+) 
(+) International Center for Mathematical Modeling in Physics and Cognitive Sciences, University of 
Växjo, S-35195 Sweden               
                                                                          and  
                                                               Joseph P. Zbilut (°) 
(°) Department of Molecular Biophysics and Physiology, Rush University,1653 W. Congress, Chicago, 
IL60612, USA. 
Abstract: For the first time we apply the methodologies of  non linear analysis to investigate atomic matter. The sense is that we use such 
methods in analysis of Atomic Weights and of Mass Number of atomic nuclei. Using AutoCorrelation Function and Mutual Information we 
establish the presence of non linear effects in the mechanism of increasing mass of atomic nuclei considered as function of the Z  atomic 
number. We also operate reconstruction in phase space and we obtain values for Lyapunov spectrum and 2D - correlation dimension. We 
find that such mechanism of increasing mass is divergent, possibly chaotic. Non integer values of 2D are found. According to previous 
studies of V. Paar et al. [ 5  ] we also investigate the possible existence of a Power Law for atomic nuclei and , using also the technique of the 
variogram, we arrive to conclude that a fractal regime could superintend to the mechanism of increasing mass for nuclei. Finally , using 
Hurst exponent, evidence is obtained that the mechanism of increasing mass in atomic nuclei is fractional Brownian regime with long range 
correlations. The most interesting results are obtained by using the Recurrence Quantification Analysis (RQA). We estimate % Recurrences, 
% Determinism, Entropy  and Max Line one time in an embedded phase space with dimension D=2 and the other time in embedding 
dimension D=1. New recurrences, psudoperiodicities, self-resemblance and class of self-similarities are identified with values of 
determinism showing oscillating values indicating the presence of more or less stability during the process of increasing mass of atomic 
nuclei All the data were analyzed using shuffled data for comparison. 
In brief, new regimes of regularities are identified for atomic nuclei that deserve to be deepened by future researches. In particular an 
accurate analysis of binding energy values by non linear methods is further required. 
Introduction 
     It is well known that the mass represents  one of the most basic properties of an atomic nucleus. 
It is also a complex and non trivial quantity whose basic properties still must be investigated deeply and 
properly understood.  
The celebrated Einstein’s mass law is  known 
m =                (1) 
On this basis some different contributions of energy  are stored inside a nucleus, and contribute to its 
mass.  
During nucleus formation in its ground state, a certain amount of energy B will be released in the process 
so that 
BcmMc
j −= ∑
22 .     (2) 
There are different sources of such energy B. It contributes  the strong attractive interaction of nucleons . 
However,  despite the immense amount of data about nuclear properties, the basic understanding of the 
nuclear strong interaction, as example, still lacks.  We have a basic model of meson exchange that of 
course works at a qualitative level but it does not provide a satisfactory approach to the description of 
such basic interaction. Still it contributes Coulomb repulsion between protons, and in addition we have 
also surface effects and still many other contributions that in a phenomenological picture are tentatively 
taken into account invoking some models as example  the liquid drop elaboration as von Weizsacker [1].  
 It is known some other  nuclear mass  models may be  considered and, despite the numerous parameters 
that are contained in these different models and the intrinsic conceptual differences adopted in their 
formulation, some common features arise from these calculations. All such models, [2], give similar 
results for the known masses, their calculations yield a typical accuracy that results about 4105 −×  for a 
medium-heavy nucleus having binding energy of the order of MeV1000 , but the predictions of such 
different mass models strongly give a net divergence when applied to unknown regions. 
One consequence of such two indications  seems rather evident. According to [2], there is the possibility 
that a basic underlying mechanism oversees the process of mass formation of atomic nuclei, and it is not 
presently incorporated and considered in the present nuclear models of the traditional nuclear physics. 
In fact, some astonishing results are not lacking  as far as this problem is concerned. 
Owing to the presence of Pauli’s exclusion principle, when nucleons are put together to form a bound 
state , there are not at rest and thus their kinetic energy also contributes to B  given in (2) and thus to the 
mass of the nucleus. Still according to [2], a part of this energy, that is to say, that one that varies 
smoothly with the number of the nucleons, is taken into account in the liquid drop model but the 
remaining part of this energy fluctuates with the number of nucleons. 
The proper nature of such fluctuations should be more investigated. 
P. Leboeuf [2] has extensively analyzed this problem and his conclusion is that  the motion of the 
nucleons inside the nucleus has  a  regular plus a  chaotic component. We will not enter into details here 
[2] but we only remember here that traditionally in nuclear physics dynamical effects in the structure of 
nuclei have been referred to as shell effects with the pioneer studies of A. Bohr and B.R. Mottelson [3]  
and V.M. Strutinsky [4]. The experience here derives from atomic physics where the symmetries of the 
Hamiltonian generates strong degeneracy of the electronic levels  and such degeneracy produce 
oscillations in the electronic binding energy. Shell effects  should be due to deviations of the single 
particle levels with respect to their average properties.  
According to the different approaches that have been introduced to reproduce the systematics of the 
observed nuclear masses that in part are inspired to liquid drop models or Thomas Fermi approximations, 
the total energy may be expressed as the sum of two contributions: 
)x,Z,N(Û)x,Z,N(U)x,Z,N(U +=          (3) 
with x  a parameter set defining the shape of the nucleus. U  is  describing the bulk or macroscopic 
properties of the nucleus and Û  instead describes shell effects. This term could be splitted in two 
components [2], the first representing the regular component and the second representing instead the 
chaotic contribution. The same thing we should have for the mass  
BcmMc
j −= ∑
with 
)x,E(B̂)x,E(B)x,E(B −=        (4) 
There is now another important  but independent contribution that deserves to be mentioned here. 
Rather recently V. Paar et al [5] introduced a power law for description of the line of stability of atomic 
nuclei, and in particular for the description of atomic weights. They compared the found power law with 
the semi-empirical formula of the liquid drop model, and showed that the power law corresponds to a 
reduction of neutron excess in superheavy nuclei with respect to the semi-empirical formula. Some fractal 
features of the nuclear valley of stability were analyzed and the value of fractal dimension was 
determined. 
It is well known that a power law may be often connected with an underlying fractal geometry of the 
considered system. If confirmed for atomic nuclei, according to [5], it could be proposed a new approach 
to the problem of stability of atomic nuclei. In this case the aim should be to identify the basic features in 
underlying dynamics giving rise to the structure of the atomic nuclei. Of course, it was pointed out the 
role of fractal geometry in quantum physics and quark dynamics [6] and in particular it was analyzed the 
self-similarity of paths of the Feynman path integral. 
Finally, M. Pitkanen repeatedly outlined that his  TGD model predicts  that universe is 4-D spin glass and 
this kind of fractal energy landscape might be present in some geometric degrees of freedom such as 
shape of nuclear  outer surface or, if nuclear string picture is accepted, in the folding dynamics of the 
nuclear string [7]. 
Still examining the problem under a different point of view, we must outline here the results that recently 
were obtained in [8]. 
These authors found non linear dynamical properties of giant monopole resonances in atomic nuclei 
investigated within the time-dependent relativistic mean field model. 
Finally, in ref [9], the statistics of the radioactive decay of heavy nuclei was the subject of experimental 
interest. It was considered that, owing to the intrinsic fluctuations of the decay rate, the counting statistics 
could depart from the simple Poissonian behaviour. Several experiments carried out with alfa and beta 
sources have found that the counting variance for long counting periods, is higher than the Poissonian 
value by more than one order of magnitude. This anomalous large variance has been taken as an 
experimental indication that the power spectrum of the decay rate fluctuations has a contribution that 
grows as the inverse of the frequency f  at low frequencies. 
In conclusion, also considered the problem from several and different view points, there are  some 
different arising evidences that it deserves to be analyzed by the methods of non linear dynamics in order 
to obtain some detailed result. 
This was precisely the aim of the present paper. We analyzed the Atomic Weights, )Z(Wa  and the Mass 
Number A(Z) as function of the atomic number Z for stable atomic nuclei and we applied to such data our 
non linear test methods, Fractal and Recurrence Quantification Analysis. The results are reported and 
discussed in detail in the following section. 
2.  Preparation of the Experimental Data. 
It is known that the trends of nuclear stability may be represented in a well known Z,N  chart of nuclides 
where each nucleus with Z protons and N  neutrons has mass Number NZA += . A line of stability may 
be realized by taking for each atomic number Z , the stable nucleus of the isotope having the largest 
relative abundance. 
The atomic weights of a naturally occurring element are  given by averaging the corresponding isotope 
weights, weighted so  to take into account the relative isotopic abundances. In this paper the data for 
stable nuclei with Z  values until 83=Z  were considered. The data were taken using the IUPAC 1997, 
standard atomic weights, at www.webelements.com. )Z(Wa    and  )Z(A   are given in Fig.1. 
Fig.1 
.             Atomic Weights: )Z(Wa       Mass Number )Z(A    
3.Tests by using Mutual Information. 
Autocorrelation Function. The autocorrelation )(τρ  is given by the correlation of a time series with itself 
using )t(x  and )t(x τ+  two time series in the correlation formula. For time series it measures as well 
correlated values of the given time series result under different delays. A  choice for the delay time to use  
when embedding time series data should be the first time the autocorrelation crosses zero. It represents 
the lowest value of delay for which the values are uncorrelated. The important thing here is that the 
autocorrelation function is a linear measure and thus it should not provide accurate results in all the 
situations in which important non linear contributions are expected. 
In the present  case we examine two series that are )Z(Wa  and )Z(A . Here we have not a time variable 
respect to which  the delay must be characterized  but instead  it is the atomic number Z that takes the 
place of time t. Therefore  we will speak here of shiftZ −  instead of time –lags in our embedding 
procedure. 
In Fig.2 we report the results of our calculations for autocorrelation function (ACF)  in the case of Atomic 
Weights and Mass Number respectively. Both the ACF for )(ZWa  and )(ZA  were calculated for 
shiftZ − ranging from 1 to 80. The first value of Z the ACF crosses the zero was obtained for )(ZWa  and 
)(ZA , and it resulted 30=− shiftZ . A typical behaviour was obtained for ACF, in the cases of   )(ZWa  
and )(ZA  respectively,  resulting in progressively, positive but decreasing values of ACF until the value 
30=− shiftZ , and a subsequent negative half-wave for shiftZ −  values greater than 30. This seems an 
interesting result that deserves in some manner a careful interpretation. 
Fig.2 
       ACF for shiftZ −  values ranging from               ACF for shiftZ −  values ranging from 1 to 80 in  
       1 to 80 in the case of  )(ZWa .                             the case of ).(ZA  
The Mutual Information.  It is usually used to determine a useful time delay for attractor reconstruction 
of a given time series. Generally speaking, we may observe only a variable from a system, )t(x , and we 
wish to reconstruct a higher dimensional attractor. We have to consider 
[ ])nt(x),.......,t(x),t(x),t(x τττ +++ 2  to produce a )n( 1+  dimensional representation. Consequently, 
the problem is to choose a proper value for the delay τ . If the delay is chosen too short, then )t(x  is very 
similar to )t(x τ+ . Of course,  for a too large delay, then the corresponding coordinates result essentially 
independent and no information can be gained. The method of Mutual Information [10] involves the idea 
that a good choice for τ   is one  that, given )(tx   provides new information with measurement  )t(x τ+ . 
In other terms, given a measurement of )(tx , how many bits on the average can be predicted about 
)( τ+tx ? In the general case, as τ  is increased, Mutual Information decreases and then usually rises again. 
The first minimum in Mutual Information is used to select a proper τ . The important thing is here that the 
Mutual Information function takes non linear correlations into account. 
Before to consider the results that we have obtained, it is important to take into account that they change 
in some manner our traditional manner to approach the discussion on atomic weights and mass numbers 
of atomic nuclei. In fact, we do not consider here values obtained for a single atomic weight or for a 
single mass number. Instead, using M.I., we evaluate M.I. values computed for pairs of Atomic Weights, 
i.e.  )(ZWa   and )( shiftZZWa −+ , for any possible Z  and for each considered .shiftZ −  The same thing 
happens for pairs of atomic nuclei with Mass Numbers )(ZA   and )( shiftZZA −+ . 
In Fig.3 we give our results for analysis of )(ZWa . The calculated shiftZ − resulted 3=Z .In Fig.4 we give 
instead the results for )(ZA . In this case the calculated shiftZ − resulted to be 2=Z . To complete our 
results, in Fig.5 we give also the results of M.I computed for )(ZN , being this time N the number of 
neutrons considered  as ).(ZN  Finally, Fig. 6 compares Mutual Information of )(ZWa , )(ZA , )(ZN . 
    Fig. 3: Mutual Information  
Z-shift M. I. 
0 2.75572 
1 2.29477 
2 2.12415 
3 2.11128 
4 2.29507 
5 2.30601 
6 2.32549 
7 2.15915 
8 2.11995 
9 2.19359 
10 2.34003 
11 2.21110 
12 2.05087 
13 2.09827 
14 2.16302 
15 2.19951 
16 2.09898 
17 2.03026 
18 2.07486 
19 2.15473 
20 2.07256 
Fig. 4: Mutual Information 
Z-shift M. I. 
0 2.75025 
1 2.24680 
2 2.12359 
3 2.12379 
4 2.24418 
5 2.25447 
6 2.21247 
7 2.15885 
8 2.11560 
9 2.19567 
10 2.25399 
11 2.09334 
12 2.02933 
13 2.08003 
14 2.13427 
15 2.07279 
16 2.09985 
17 1.98171 
18 2.08486 
19 2.11095 
20 2.09009 
Mutual Information Atomic Weights
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Z(lags)
Mutual Information Mass Number
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Z(lags)
Fig. 5: Mutual Information 
Z-shift M.I. 
0 2.746829 
1 2.247466 
2 2.117802 
3 2.108233 
4 2.248289 
5 2.251926 
6 2.288034 
7 2.144352 
8 2.131585 
9 2.133403 
10 2.219288 
11 2.086516 
12 2.082935 
13 2.129288 
14 2.226088 
15 2.092718 
16 2.034579 
17 1.983242 
18 2.053253 
19 2.038391 
20 2.018196 
Fig.6 : Mutual Information. 
Mutual Information 
Atomic Weights-Mass Number-Neutron Number
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Z-shift
atomic w eights
mass number
neutron number
We are now in the condition to reassume some results. 
Using autocorrelation function, ACF (Linear Analysis), a shiftZ −  value of 30=Z  is obtained for both 
)(ZW a  and )(ZA . 
Using Mutual Information (Non Linear Analysis) it is obtained instead 3=− shiftZ  for )(ZWa   and 
2=− shiftZ  for )(ZA . Also )(ZN  gave 3=− shiftZ . 
We have a preliminary indication that the mechanism of increasing mass in atomic nuclei is a non linear 
mechanism. Of course, this could be an important indication in understanding of the basic features of  
nuclear matter .Therefore  it becomes of relevant importance to attempt to confirm such conclusion on the 
Mutual Information Neutron Number
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Z-shift
basis of a more deepened control. The test that in such cases one uses in analysis of non linear dynamics 
of time series data is that one of  surrogate data. Here we  used shuffled data. The results are given in 
Fig.7 for )(ZWa  and in Fig.8 for )(ZA . 
Fig.7 : Surrogate Data Analysis 
Z-lags M.I.-Surrogate Data 
0 2.76156 
1 1.48621 
2 1.44105 
3 1.44755 
4 1.38917 
5 1.36923 
6 1.54505 
7 1.38976 
8 1.38138 
9 1.48849 
10 1.36547 
11 1.34689 
12 1.37347 
13 1.42643 
14 1.34143 
15 1.29609 
16 1.25763 
17 1.40470 
18 1.43727 
19 1.41390 
20 1.36288 
Fig. 8 : Surrogate Data Analysis 
Z-lags M.I.-Surrogate Data
0 2.73606 
1 1.61453 
2 1.39896 
3 1.37041 
4 1.33673 
5 1.30566 
6 1.41145 
7 1.34616 
8 1.35618 
9 1.34365 
10 1.37382 
11 1.25994 
12 1.30270 
13 1.41078 
14 1.47545 
15 1.21417 
16 1.31047 
17 1.27582 
18 1.38925 
19 1.35851 
20 1.41464 
The results obtained by using shuffled data clearly confirm that we are in presence of a on linear 
mechanism in the  process of increasing mass of atomic nuclei. 
We also tested statistically  the obtained differences between M.I. for  original and surrogate data for the 
case of Atomic Weights as well as for the case of Mass Numbers. In the case of M.I for  Atomic Weights 
vs M.I. Atomic Weights – Surrogate Data , using Unpaired t test we obtained a  P value P<0.0001 and the 
same value was found in the case of M.I. Mass Number vs M.I. Mass Number – Surrogate Data. 
Mutual Information 
Atomic Weights vs Surrogate Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Z(lags)
atomic w eigths
surrogate data
Mutual Information 
Mass Number vs Surrogate Data
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Z(lags)
mass number
surrogate data
In conclusion, by accepting the presence of non linearity, we have reached  the first relevant conclusion of 
the present paper. 
 Looking now to Figures 3, 4, 5, 6 one may identify now new properties for atomic nuclei. Remember that 
we are considering each time, pairs of Atomic Weights or pairs of Mass Numbers or still pairs of Neutron 
Numbers for atomic nuclei with shiftedZ −  values ranging from 1 to 20.  What one should expect in this 
case is to find a minimum of  Mutual Information followed soon after by a rather constant behaviour for 
M.I. Examination of the results reveals instead that we have  some definite maxima  and some definite 
minima at given values of shiftZ −  that are quite different in the two and three cases that we have 
examined. 
In detail the maxima for Atomic Weights are given at Z-shift values = 6,10,15,19,….   . Minima instead 
are given at Z-shift values =3,8,12,17,…… . 
The maxima for Mass Numbers are given at Z-shift values=5,10,14,16(19)  while the minima are given at 
Z-shift values =2,8,12,15,17. For Neutrons we have Z-shift values = 3,9,12,17 for the minima and 
6,10,14,18 for the maxima. In conclusion: Still repeating here  that each time we are exploring the M.I 
value for pairs of atomic weights, or of mass number or of neutrons, shifted in the valueZ −  by some 
given values ranging between 1 and 20, we find that some pairs of nuclei show maxima MI values while 
other pairs of nuclei show minima MI value. Therefore  we have new and  interesting properties identified 
in atomic nuclei when analyzed by pairs as in the present methodology.  We may call such new identified 
regularities for atomic nuclei as pseudo periodicities in pairs of atomic nuclei.  
For Mass Numbers we may write as example that 
shiftZNA −+∆=∆  
Fixed  a value of Z , we have consequently 11 NZA += . For an atomic nucleus with muss number 2A  and 
Z-shifted, we will have 22 NshiftZZA +−+= . 
Consequently, shiftZNAAA −+∆=−=∆ 12 with 12 NNN −=∆ . For  Z-shift values=5,10,14,16(19), the 
considered pairs of atomic nuclei will show maxima of M.I.. Instead for  Z-shift values =2,8,12,15,17, 
such M.I. values will reach a minimum value. 
Let us go in more detail in the analysis. 
First of all we have also to note that  the values of MI, calculated for each shiftZ − ranging from 1 to 20,  
result to be quite different in )(ZWa  respect to )(ZA , and N(Z). In addition, as previously said, Mutual 
Information measures how much, given two random variables, and knowing one of these two variables, is 
reduced our uncertainty about the other. Mutual Information must thus be intended essentially as 
estimation of mutual dependence of two variables. In our case we find that the pairs of atomic weights, or 
of mass number or of neutrons in  atomic nuclei that are shifted by some definite values of the atomic 
number Z , show strong dependence (maxima values of M.I.) or, respectively, they show  very low 
dependence (minima values of M.I.). We have some new pseudoperiodicities that in some manner recall  
a new kind of pseudo isotopies. All that seems to be realized in a full regime of non linearity. 
4. Phase Space Reconstruction of )(ZWa  and )(ZA . 
We may now attempt to obtain  for the first time a phase space reconstruction of Atomic Weights and 
Mass Number of atomic nuclei. 
To reach this objective one must estimate Embedding Dimension using the False Nearest Neighbors 
Criterion (FFN). We applied it using a shiftZ −  = 3 for )(ZWa  and a 2=− shiftZ  for )(ZA  as previously 
found. A false criterion distance was considered to be 4.42 for both the cases of the analysis. The results 
are reported in Fig.9 for atomic weights, )(ZWa , and in Fig.10 for Mass Number, )(ZA . 
Fig. 9 :False Nearest Neighbors for Atomic  Fig.10 : False Nearest Neighbors for Mass  
 Weights        Number 
The evaluation of the results given in Figures 9, 10 enables us to conclude that the phase space 
reconstruction for atomic weights and mass number requires an estimated embedding dimension ,that 
results to be included between 1 and 2. We may assume to consider 2=D . Atomic weights and mass 
number of atomic nuclei may be approximately represented in a bi dimensional phase space. 
Consequently, according to the general framework of the  theory on non linear dynamics of  systems, we 
may conclude that a very few number of independent variables is required in order to describe the 
mechanism of increasing mass of atomic nuclei. We may accept to consider that  they are two variables 
that, with greatest prudence, we may accept to identify as being  the proton and the neutron numbers, 
respectively. The phase space description of atomic weights )(ZWa and of Mass Number )(ZA  requires 
with approximation, the use of such two variables. 
 Since this result has been obtained in a closed form, we may now attempt to analyze if the two given 
)(ZWa  and )(ZA  exhibit or not properties of divergence. 
To this purpose we may calculate Lyapunov spectrum in the embedded phase space. The results that we 
obtained, are reported in Fig. 11 and in Fig.12 for )(ZWa  and )(ZA , respectively. 
Fig.11 : Lyapunov spectrum of atomic weights 
iteration, exponents  
1 -0.139132 -1.767928 
2 -0.054642 -1.852418 
3 -0.032002 -1.875058 
4 -0.021497 -1.885562 
iteration, exponents  
5 -0.015293 -1.891767 
6 -0.011169 -1.895891 
7 -0.008224 -1.898835 
8 -0.006016 -1.901043 
iteration, exponents  
9 -0.004299 -1.902761 
10 -0.002925 -1.904134 
11 -0.001801 -1.905258 
12 -0.000864 -1.906195 
13 -0.000946 -1.854474 
14 -0.000219 -1.810264 
15 0.000026 -1.779827 
16 0.000273 -1.753227 
17 0.000425 -1.705629 
18 0.000914 -1.667297 
19 0.001349 -1.632997 
20 0.001738 -1.602124 
21 0.002186 -1.585933 
22 0.002567 -1.570617 
23 0.002785 -1.552573 
24 0.003007 -1.536054 
25 0.003233 -1.508474 
26 0.003433 -1.483006 
27 0.003304 -1.450959 
28 0.003100 -1.421118 
29 0.003155 -1.394447 
30 0.002734 -1.365179 
31 0.002481 -1.336786 
32 0.002367 -1.309490 
33 0.002297 -1.283733 
34 0.002104 -1.261030 
35 0.002071 -1.235585 
36 0.002229 -1.208038 
37 0.002491 -1.182091 
38 0.002558 -1.161598 
39 0.002648 -1.143582 
40 0.002781 -1.125369 
41 0.002912 -1.108048 
42 0.003353 -1.094097 
43 0.003464 -1.078832 
44 0.003659 -1.064133 
iteration, exponents  
45 0.003833 -1.043353 
46 0.003724 -1.022886 
47 0.003713 -1.008188 
48 0.003649 -0.995043 
49 0.003576 -0.982424 
50 0.003419 -0.970124 
51 0.003116 -0.958012 
52 0.002781 -0.948094 
53 0.002434 -0.938526 
54 0.002281 -0.929092 
55 0.002133 -0.920000 
56 0.001982 -0.911222 
57 0.001851 -0.902767 
58 0.001885 -0.897098 
59 0.001945 -0.891648 
60 0.002004 -0.886751 
61 0.002097 -0.882050 
62 0.002113 -0.877724 
63 0.002311 -0.872891 
64 0.002398 -0.865438 
65 0.002469 -0.858201 
66 0.002539 -0.854135 
67 0.002586 -0.850170 
68 0.002372 -0.850370 
69 0.002156 -0.850555 
70 0.001942 -0.850731 
71 0.001733 -0.850900 
72 0.001529 -0.851065 
73 0.001330 -0.851224 
74 0.001136 -0.851379 
75 0.000947 -0.851530 
76 0.000763 -0.851676 
77 0.000585 -0.851819 
78 0.000410 -0.851958 
Fig.12 : Lyapunov spectrum of mass number 
iteration, exponents                iteration,          exponents 
1 -0.063750 -1.815136 
2 -0.013655 -1.865231 
3 -0.004496 -1.874390 
4 -0.000940 -1.877946 
5 0.001067 -1.879953 
6 0.002390 -1.881276 
iteration, exponents 
7 0.003333 -1.882218 
8 0.004039 -1.882925 
9 0.004589 -1.883475 
10 0.005029 -1.883914 
11 0.005907 -2.030549 
12 0.005902 -2.152007 
13 0.007184 -2.344940 
14 0.007900 -2.508600 
15 0.008538 -2.585360 
16 0.008446 -2.593648 
17 0.008353 -2.600949 
18 0.007751 -2.585976 
19 0.008750 -2.554168 
20 0.008150 -2.531907 
21 0.008186 -2.482481 
22 0.007894 -2.450329 
23 0.008806 -2.420931 
24 0.009633 -2.393973 
25 0.009816 -2.350399 
26 0.009954 -2.310147 
27 0.009880 -2.258690 
28 0.009662 -2.210759 
29 0.009018 -2.163469 
30 0.008250 -2.119600 
31 0.007832 -2.082773 
32 0.007498 -2.052309 
33 0.007109 -2.023616 
34 0.006523 -2.013469 
35 0.005691 -2.003623 
36 0.005143 -1.995782 
37 0.004718 -1.988460 
38 0.003966 -1.963859 
39 0.004116 -1.946858 
40 0.004339 -1.930786 
41 0.004649 -1.922343 
42 0.005367 -1.917894 
43 0.007274 -1.903393 
iteration, exponents 
44 0.008790 -1.889480 
45 0.010447 -1.876392 
46 0.010388 -1.871786 
47 0.009970 -1.869670 
48 0.009374 -1.855670 
49 0.008541 -1.841986 
50 0.007498 -1.829730 
51 0.006726 -1.821934 
52 0.005940 -1.808215 
53 0.005215 -1.795046 
54 0.004497 -1.783516 
55 0.004030 -1.771905 
56 0.003596 -1.760725 
57 0.002953 -1.750916 
58 0.002916 -1.743006 
59 0.002961 -1.735446 
60 0.003004 -1.728808 
61 0.003032 -1.722373 
62 0.003108 -1.713120 
63 0.003229 -1.704206 
64 0.003291 -1.693598 
65 0.003299 -1.689223 
66 0.003127 -1.691107 
67 0.002824 -1.702828 
68 0.002620 -1.700615 
69 0.002292 -1.708755 
70 0.002307 -1.709571 
71 0.002341 -1.710383 
72 0.002445 -1.714966 
73 0.002536 -1.719414 
74 0.002212 -1.719409 
75 0.001877 -1.719385 
76 0.001547 -1.719358 
77 0.001225 -1.719332 
78 0.000912 -1.719306 
79 0.000606 -1.719280 
80 0.000308 -1.719255 
To calculate the Lyapunov spectrum in the case of the Atomic Weights, )(ZWa , we used  a number of 22 
fitted points in the embedded phase space. 
These are the following results for the calculated Lyapunov exponents:  
λ1 = 0.000410 and λ2 = -0.851958. It is seen that we have 01 >λ   and 02 <λ  with 021 <λ+λ  as required. 
In conclusion we are in presence of a divergent system and such divergence may be indicative a pure 
chaotic regime for Atomic Weights. 
In the case of the Mass Number, )(ZA , we utilized a number of 17 fitted points in the embedded phase 
space. These are the results we obtained for the calculated Lyapunov exponents: 000308.01 =λ , 
719255.12 −=λ   with 01 >λ  , 02 <λ  and 021 <λ+λ . Also in the case of Mass Number, )(ZA ,  we are in 
presence of a divergent system and it could be indicative of a pure chaotic regime. 
In brief, we have reached the following conclusions: 
1) In the process of increasing mass of atomic nuclei we are in presence of a non linear mechanism. 
Remember that the presence of   non linear contributions in the dynamic of a system gives  often 
origin to  chaotic regimes. 
2) The mechanism of increasing mass in Atomic Weights and  in  Mass Number for pairs of atomic 
nuclei also exhibits some pseudo periodicities at some definite shiftsZ − of atomic nuclei. 
Therefore, we could be in presence of  an ordered  regime of increasing mass but  in the complex 
of a whole structure that is divergent and possibly chaotic. 
3) A phase space reconstruction has been realized for Atomic Weights and Mass Number of atomic 
nuclei, respectively. In our opinion this is a relevant result that is obtained here. In fact, from the 
analysis performed by using F.F.N, it does not result in a so clear manner that the reconstructed  
phase space has dimension D=2. We have F.F.N. values that suspend embedding dimension 
between 1 and 2. In this case in the analysis it is adopted the greatest value. In conclusion we may 
accept an embedding dimension D=2, and only in this condition we have that only few, two 
variables, are required in order to describe the mechanism of increasing mass of atomic nuclei in 
phase space with respect to atomic weights and mass number . The first variable should be the 
atomic number, Z , and the other variable should be the Neutron Number , .ZAN −=  
4) The analysis of )(ZWa and )(ZA  reveals a new important features when we analyze such two 
systems by calculation of Lyapunov spectrum. It results that we are in presence of divergent 
systems in both case of stable nuclei analyzed by )(ZWa  and )(ZA . Such divergent property, 
linked to the previously found on non linearity, could be indicative that we are in presence of a 
chaotic regime in the  mechanism of the increasing mass of atomic  nuclei  when  seen as function 
of Z . 
We may go on by a further step calculating the Correlation Dimension in the reconstructed phase space 
for both )(ZWa  and )(ZA . In Fig.13 we report the results for )(ZWa . In Fig. 14 we give instead the results 
for )(ZA . Finally in Figures 15 and 16 we have the results using surrogate (shuffled) data. 
Fig.13 : Atomic Weights. 
Fig.14 : Mass Number. 
For the atomic weights it is obtained  D2 = 1.955 ± 0.296 as value for Correlation Dimension. For Mass 
Number it results instead  D2 = 2.120 ± 0.084. 
It is important to observe that in both case we obtain non integer values of such topological dimension in 
phase space reconstruction. 
We may now consider the results for surrogate data. 
Fig.15 : Results on Correlation Dimension, Atomic Weights (surrogate data). 
Fig.16 : Results on Correlation Dimension, Mass Number (surrogate data).  
In the case of )(ZWa  we obtain D2 = 5.130 ± 0.624  while instead in the case of )(ZA  we have D2 = 5.193 
± 0.810 
As seen through the results, a net difference is obtained in comparison of original with surrogate data. 
They may be quantified in the following manner: 
)2(793.3
),4(088.5),2(678.3
=−=−=
lagZNumberMass
lagZWeightsAtomiclagZWeightsAtomic
surrogate
surrogateoriginal
The null hypothesis may be rejected. In conclusion we have a non integer topological dimension for both 
)(ZWa  and ).(ZA  
Therefore it has a sense to attempt to ascertain if we are in presence of a fractal behaviour for both the 
system of data that we have in examination. 
5. On a Possible Existing Power Law to Represent Increasing Mass in Atomic Weights and Mass 
Number of Atomic Nuclei. 
As we indicated in the introduction in the present paper, rather recently V. Paar et al [6] introduced a 
power law for description of the line of stability of atomic nuclei, and in particular for the description of 
atomic weights. They compared the found power law with the semi-empirical formula of the liquid drop 
model, and showed that the power law corresponds to a reduction of neutron excess in superheavy nuclei 
with respect to the semi-empirical formula. Some fractal features of the nuclear valley of stability were 
analyzed and the value of fractal dimension was determined. 
It is well known that a power law may be often connected with an underlying fractal geometry of the 
considered system. If confirmed for atomic nuclei, according to [5], it could be proposed a new approach 
to the problem of stability of atomic nuclei. In this case the aim should be to identify the basic features in 
underlying dynamics giving rise to the structure of the atomic nuclei. 
The aim is to perform here such kind of analysis for )(ZWa  and )(ZA . 
For atomic weights let us introduce the following Power Law : 
β= aZZWa )(                            (5) 
while instead for the Mass Number let us introduce the following power law 
γ= cZZA )(                                    (6) 
The problem is now to estimate ( β,a )  and  ( γ,c ) by a fitting procedure. 
We give here the obtained results for the Atomic Weights. 
Curve Fit Report 
Y Variable: C2.  X Variable: C1. Model Fit:   A*X^B  
Parameter Estimates for All Groups 
Groups CountIter's            R2                               A                             B  
All 83 24 0.99954 1.47335 1.12133  
Combined Plot Section 
Fig. 17 
125.0
187.5
250.0
0.0 25.0 50.0 75.0 100.0
Y = A*X^B
                     
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 1.47335 0.02448 1.42464 1.52206 
B 1.12133 0.00399 1.11338 1.12927 
Iterations 24 Rows Read 83 
R-Squared 0.999538 Rows Used 83 
Random Seed 9839 Total Count 83 
Estimated Model 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Plot Section 
           
Fig.18 
0.0 25.0 50.0 75.0 100.0
Residual vs C1
In conclusion, for the (5) we obtained  a = 1.47335  and β = 1.12133. 
V. Paar et al. [5] obtained  instead that a =1.44±0.02  and β=1.120±0.004 and β=1.19±0.01 by using  the 
Box Counting method. There is an excellent agreement. 
As it may be seen the obtained values significantly differ from the line. In addition, the obtained values 
strongly give evidence for a  possible fractal regime. 
Let us see now the results that we obtained for the (6) concerning Mass Number. 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit:   A*X^B  
Parameter Estimates for All Groups 
Groups CountIter's             R2                              A                               B  
All 83 23 0.99929 1.46185 1.12389  
Combined Plot Section 
Fig.19 
125.0
187.5
250.0
0.0 25.0 50.0 75.0 100.0
Y = A*X^B
                      
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 1.46185 0.03010 1.40195 1.52174 
B 1.12389 0.00495 1.11404 1.13373 
Iterations 23 Rows Read 83 
R-Squared 0.999294 Rows Used 83 
Random Seed 10007 Total Count 83 
Estimated Model 
 Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Plot Section 
Fig.20 
0.0 25.0 50.0 75.0 100.0
Residual vs C1
            
In conclusion,  for the (6)  we obtained: c=1.46185, γ=1.12389.  
V. Paar et al. obtained [6] c=1.47±0.02 and γ=1.123±0.005 in excellent agreement. 
Again we may conclude for values that significantly differ from line. In addition, the obtained values 
strongly give evidence for a  possible fractal regime. 
The possible existence of a fractal regime in the mechanism of increasing mass in atomic weights and 
mass umber of atomic nuclei changes radically our traditional manner to conceive nuclear matter. 
Consequently, it becomes of relevant importance to attempt to deepen such result so  to reach the highest 
possible level of certainty on it. A way to deepen such kind of analysis is to follow the way of variogram 
method. Variograms usually give powerful indications on the variability of the examined data, on their 
self-similarity and self-affine behaviour. In particular, they enable us to calculate the Generalized Fractal 
Dimension [ for details see ref.11]. 
The semivariogram is given in the following manner 
))()((
2hxRxRE
=γ        (7) 
For Atomic Weights it is: 
))()((
2hZWZWE
=γ       (8) 
and for Mass Number it is 
))()((
2hZMZME
=γ   .  shiftZ − is indicated here by  ,.....2,1=h   .  (9) 
Still, in the general case we may write 
ii xRhxRhN
2))()((
)(  (10) 
For a self-affine series the semivariogram scales according to   
DhCh =γ )(      (11) 
being D the Generalized Fractal Dimension. It is linked to aH  by aHD 2= being aH the Hausdorff 
dimension. 
We may also estimate the corresponding Probability Density Function that is given in the following 
manner 
1)( −= a
hP       (12) 
being  1−= aD   and k is a scale parameter. 
Let us introduce now the results that we obtained for Atomic Weights. 
RESULTS 
Fig. 21: Variogram of atomic weights 
     
Z-shift     value      Z-shift     value       Z-shift       value     
1 4.1038233 28 2583.4415 55 9936.2738
2 13.890723 29 2774.3962 56 10263.722
3 30.73476 30 2974.6827 57 10614.884
4 53.281165 31 3177.0212 58 10974.746
5 82.706811 32 3391.3496 59 11360.41 
6 118.39056 33 3607.1587 60 11745.812
7 161.00905 34 3826.8629 61 12160.26 
8 209.83439 35 4053.0824 62 12559.606
9 265.1562 36 4283.8682 63 12969.082
10 326.91086 37 4517.609 64 13342.27 
11 395.38036 38 4760.8117 65 13738.976
12 470.35989 39 5010.0734 66 14163.507
13 551.85195 40 5268.4938 67 14573.838
14 640.44287 41 5533.4834 68 14979.658
15 735.22246 42 5805.6797 69 15415.604
16 837.44107 43 6083.8624 70 15830.968
17 946.67776 44 6370.1678 71 16277.052
18 1060.8571 45 6665.5819 72 16709.305
19 1183.535 46 6969.6876 73 17165.872
20 1314.6237 47 7285.3773 74 17613.299
21 1448.8492 48 7605.9824 75 18096.431
22 1590.1856 49 7928.8907 76 18533.473
23 1734.6391 50 8261.1897 77 18994.677
24 1888.6115 51 8592.0056 78 19457.937
25 2047.654 52 8917.3226 79 20009.547
26 2217.1471 53 9259.8625 80 20578.44 
27 2394.576 54 9591.9184
Using the (11) in a Ln-Ln representation, we may now estimate the Generalized Fractal Dimension. We 
obtained the following results. 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Parameter Estimates for All Groups 
Groups            Count     Iter's R2 A B  
All 81 6           0.99986                    1.25542                 1.98086  
Combined Plot Section: Ln-Ln variogram fitting 
Fig.22 
0.0 1.3 2.5 3.8 5.0
Y = Simple Linear
                      
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 1.25542 0.00939 1.23674 1.27410 
B 1.98086 0.00264 1.97560 1.98612 
Iterations 6 Rows Read 81 
R-Squared 0.999859 Rows Used 81 
Random Seed 7364 Total Count 81 
Estimated Model 
(1.25542232430747)+(1.98086225009967)*(C1) 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Plot Section 
Fig.23 
0.0 1.3 2.5 3.8 5.0
Residual vs C1
             
In conclusion, we obtain for Atomic Weights the following results: 
Generalized Fractal Dimension  D = 1.98086 
Hausdorff dimension     Ha = 0.99043. 
By using the (12) we may now calculate the Probability Density Function. For atomic weights it results 
that  
P(Z) = 98086.1610673.5 Z−×            (13) 
that is given in Fig.24. 
Fig. 24 
In order to deepen our analysis we may  also employ  a modified version of standard variogram analysis, 
using this time  a light modification of its usual form in the following way  
iiN hxRxRN
2 ))()((2
)(     (14) 
where we calculate  now by )2/1( N instead of )(2/1 hN −  being )1(2 −N the number of degrees of freedom 
for the whole system taken in consideration. 
We have  the results in the case of the variogram N2γ  for atomic weights in Figures 25, 26, 27. 
Fig. 25 
Fig.26 
Fig.27    
As see, passing from variogram in Fig.25 to variogram in Fig.26 and, finally, in variogram in Fig.27 we 
have used each time a different factor of scale and, in spite of such different factors of scale, the behaviour 
of the correspondent variograms, remain unchanged. This result may be taken as further indication that we 
are in presence of a fractal regime. 
In addition, by  the N2γγγγ variogram, we may now re-calculate the Generalized Fractal Dimension using a 
Ln-Ln scale.  
The results are given in the following scheme. 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Parameter Estimates for All Groups 
Groups Count Iter's R2 A B  
All 52 6          0.99288                   1.65610                  1.70683  
Combined Plot Section 
Fig.28 
0.0 1.0 2.0 3.0 4.0
Y = Simple Linear
                      
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 1.65610 0.06408 1.52739 1.78481 
B 1.70683 0.02045 1.66576 1.74790 
Iterations 6 Rows Read 52 
R-Squared 0.992875 Rows Used 52 
Random Seed 10882 Total Count 52 
Estimated Model 
(1.65609637745411)+(1.70683194557246)*(C1) 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Plot Section 
Fig.29 
0.0 1.0 2.0 3.0 4.0
Residual vs C1
             
In conclusion, also in this case a non integer  value of the Generalized Fractal Dimension is obtained. It 
results D=1.70683 with Hausdorff dimension 853415.0=aH . Such values result in satisfactory accord 
with those previously had in the case of the standard variogram. 
We may now consider the results that we obtained in the corresponding analysis for Mass Number. 
Fig. 30:Variogram of Mass Number 
Variogram values: 
 Z-lags      Variogram- value              Z-lags      Variogram- value
1 5.243902 
2 14.7284 
3 32.275 
4 54.56329 
5 84.44231 
6 119.9221 
7 163.0592 
8 212.2067 
9 268.1824 
10 329.637 
11 399.0625 
12 474.9789 
13 557.2214 
14 646.1667 
15 741.9779 
16 843.4851 
17 953.9318 
18 1068.915 
19 1192.695 
20 1324.532 
21 1460.766 
22 1604.451 
23 1749.508 
24 1903.856 
25 2063.957 
26 2234.096 
27 2412.473 
28 2603.482 
29 2796.472 
30 3000.292 
31 3205.394 
32 3423.01 
33 3639.03 
34 3860.969 
35 4091.052 
36 4326.67 
37 4561.033 
38 4804.122 
39 5055.284 
40 5320.965 
41 5591.036 
42 5867.11 
43 6145.663 
44 6428.603 
45 6726.934 
46 7036.689 
47 7356.458 
48 7675.757 
49 7997.926 
50 8333.5 
51 8671.016 
52 8996.565 
53 9339.117 
54 9664.879 
55 10009.8 
56 10334.06 
57 10688.52 
58 11053.98 
59 11444.02 
60 11836.33 
61 12256.52 
62 12651.02 
63 13058.83 
64 13431.24 
65 13832.53 
66 14249.85 
67 14660.66 
68 15086.9 
69 15531.39 
70 15943.19 
71 16399.63 
72 16814.68 
73 17282.35 
74 17737.39 
75 18219.06 
76 18626.5 
77 19078.67 
78 19562.9 
79 20150.38 
80 20672.67 
81 21218.5 
We may now give the estimation of the Generalized Fractal Dimension.      
Curve Fit Report (Ln-Ln plot) 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Parameter Estimates for All Groups 
Groups Count Iter's R2 A B  
All 81 4          0.99943                   1.32691                   1.96370  
Combined Plot Section 
Fig.31 
0.0 1.3 2.5 3.8 5.0
Y = Simple Linear
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 1.32691 0.01877 1.28955 1.36427 
B 1.96370 0.00528 1.95318 1.97422 
Iterations 4 Rows Read 81 
R-Squared 0.999428 Rows Used 81 
Random Seed 2960 Total Count 81 
Estimated Model 
(1.32690928509202)+(1.96369892532774)*(C1) 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Plot Section 
Fig.32      
0.0 1.3 2.5 3.8 5.0
Residual vs C1
The analysis enables us to give the following results: 
Generalized Fractal Dimension  D = 1.96370 
Hausdorff dimension     Ha = 0.98185 
We may now calculate the Probability Density Function. It assumes the following form 
P(Z) = 9637.1610786.6 Z−−−−××××  
Fig. 33 
Let us proceed estimating N2γ  at different  scale factors. 
Fig. 34 
  Fig. 35a 
Fig. 35b 
           
By the N2γ variogram  we may now re-calculate the Generalized Fractal Dimension using a Ln-Ln scale. The 
results are given in the following scheme. 
 Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Parameter Estimates for All Groups 
Groups Count Iter's R2 A B  
All 49 4           0.99538                   6.81261                  1.70270  
Combined Plot Section 
Fig.36 
                  
0.0 1.0 2.0 3.0 4.0
Y = Simple Linear
                          
Model Estimation Section 
Parameter Parameter Asymptotic Lower Upper 
Name Estimate Standard Error 95% C.L. 95% C.L. 
A 6.81261 0.05209 6.70783 6.91739 
B 1.70270 0.01692 1.66867 1.73674 
Iterations 4 Rows Read 49 
R-Squared 0.995381 Rows Used 49 
Random Seed 11153 Total Count 49 
Estimated Model 
(6.81260680697216)+(1.70270493930648)*(C1) 
Curve Fit Report 
Y Variable: C2.  X Variable: C1.  
Model Fit: C2=A+B*(C1)  Simple Linear  
Plot Section 
Fig.37 
0.0 1.0 2.0 3.0 4.0
Residual vs C1
          
Conclusion: also in this case a non integer  value of the Generalized Fractal Dimension is obtained. It 
results D=1.70270 with Hausdorff dimension 85135.0=aH . Such values result in satisfactory accord with 
those previously obtained in the case of the standard variogram. 
In conclusion, until here we have used  the standard methodologies that  generally one utilizes with the 
aim to ascertain the presence of non linear contributions in the investigated dynamics as well as to 
reconstruct phase space dynamics and to evaluate the possible presence of divergent features in the 
system, possibly of chaotic nature, and still the probable presence of a fractal regime in such dynamics. 
On the basis of the results that we have obtained, it seems very difficult to escape the conclusion that the 
process of increasing mass, regarding Atomic Weighs and Mass Number in atomic nuclei, concerns all the 
basic features of non linearity, divergence, possible chaoticity and fractality that we have only just 
indicated for systems with non linear dynamics. This is a conclusion that in some manner overthrows our 
traditional manner to approach nuclear matter. For this reason it requires still more detailed deepening. In 
the following sections we will support our conclusion by other detailed results. 
6. Calculation of Hurst Exponent and Possible Presence of Fractional Brownian Behaviour In 
Atomic Weights and Mass Number of Atomic Nuclei 
It is known that time series arise often from a random walk  usually  called Brownian motion. The Hurst 
exponent [12] in such cases is calculated to be 0.5. 
This concept may be generalized introducing the Fractional Brownian Motion (fBM) which arises from 
integrating correlated –coloured noise. 
The value of Hurst exponent helps us to identify the nature of the regime we have under examination. In 
detail, if the H exponent results greater than 0.5, we are in presence of persistence, that is to say, past 
trends also persist into the future. On the other hand, in presence of H exponent values less than 0.5 we 
conclude for anti persistence, indicating it in this case  that past trends tend to reverse in the future. 
In the present case the analysis is not performed having a time series but instead  we consider the atomic 
number Z in )(ZWa , the atomic weights, and in )(ZA , the Mass Number of atomic nuclei. 
Our analysis gave the following results. 
For atomic weights, )(ZWa , we obtained the subsequent value: 
Hurst exponent H = 0.9485604 ; SDH = 0.00625887 ; r2 = 0.999645 
Instead for Mass Number , )(ZA , we had the next value: 
Hurst exponent H = 0.8953571 ; SDH = 0.0057648 ; r2 = 0.999753. 
Both the results obtained respectively for Atomic Weights and for Mass Number, enable us to conclude 
that: 
1) we are in presence of a Fractional Brownian Regime in both the cases; 
2) in both )(ZWa  and )(ZA  the tendency is for the persistence that results more marked in )(ZWa  respect 
to )(ZA ; 
3) in the case of the Atomic Weights, )(ZWa , the value of Fractal Dimension results to be 
      0514396.12 =−= HD   
      while in the case of  Mass Number, )(ZA , the value of  Fractal Dimension is  
      1046429.12 =−= HD  . 
7. Recurrence Quantification Analysis – RQA 
Further important information on the nature of the processes presiding over the mechanism of increasing 
mass in Atomic Weights and Mass Number of atomic nuclei may be obtained by using RQA, the 
Recurrence Quantification Analysis. 
This is a kind of analysis that, as it is well known, was introduced by J.P Zbilut and C.L. Webber [13]. 
Such investigation offers a new opportunity to us. By it we may give a look to the process of increasing 
mass of atomic nuclei analyzing in detail the kind of dynamics that governs such mechanism. Therefore, 
the results of such investigation must be considered with particular attention owing to their relevance. 
The features that we may investigate in detail are the following: first of all we may evaluate the level of 
recurrence, that is to say of “periodicity”, that such process exhibits. This is obtained by estimating the % 
Rec in RQA. Soon after we may also calculate the Determinism that is involved in such process. This is to 
say that we evaluate the level of predictability that it has. We estimate such features by %Det. in RQA. As 
third RQA variable we may also estimate the entropy and than the Max Line that is a measure linked 
proportionally to the inverse of the Lyapunov exponent. In brief, such measure enables us to evaluate still 
again the possible divergence involved in such mechanism. 
Usually, when using RQA, one starts with an embedding procedure of the given time series and thus 
providing with a given reconstruction of the given time series in phase space. In our case such 
reconstruction was previously performed in previous sections and we obtained  that we should use an 
embedding dimension 2=D  with a 3=− shiftZ  in the case of Atomic Weights and a 2=− shiftZ  in the 
case of Mass Numbers. However, in the present analysis our purpose is slightly different in the sense that 
we aim to preserve the embedded dimension 2=D  but we yearn for analyzing the behaviour of the basic 
RQA variables as %Rec., %Det., ENT., and Max Line shifting step by step the value of the atomic 
number Z  so to explore the mechanism as well as Z increases step by step. In order to perform such kind 
of analysis  a  value of the distance R  should be correctly selected. Usually, the distance R  in RQA may 
be fixed rather empirically selecting a proper value so that %Rec. remain about 1%.  However Zbilut and 
Webber [13] in their RQA software package introduced RQS that estimates recurrences at various 
distances and the cut off that one has at a particular distance respect to a flat behaviour. In this manner one 
selects the best optimized distance R to use in the analysis. We applied RQS software to select the proper 
distance and it was obtained that such value should be taken 4=R  
We also ascertained that such selected value remained rather constant when increasing Z  step by step.  In 
Fig. 38 we give some of the results that were obtained.  
Fig. 38 
Estimation of  Recurrences at various Distances
(Ln-Ln Plot)
0 1 2 3 4 5
Distance R
Optimized distance R = 4 
Atomic Weights
Estimation of  Recurrences at various  Distances
(Ln-Ln Plot)
0 0.5 1 1.5 2 2.5 3 3.5
Distance R
Optimized distance R = 4 
Mass Number
In conclusion we selected R=4 for the distance to use in RQA. The embedding dimension was chosen to 
be D=2 as it resulted by using FNN criterion and verifying this choice also for different Z  values. 
Finally, we decided to use the value  L=3 for the Line Length. 
We have obtained the following results. 
Recurrence Quantification Analysis applied to Atomic Weights, )(ZWa , for increasing values of shiftZ − . 
The results obtained for %Rec., %Det., ENT., and MaxLine are reported in the following Table 1 
Table 1 
Z-shift %Rec. %Det. Entropy Max-Line
1 1.36 73.33 2.00 14 
2 1.39 44.44 1.50 7 
3 1.33 52.38 1.59 13 
4 1.36 28.57 1.00 7 
5 1.30 38.46 1.00 11 
6 1.37 27.50 0.92 5 
7 1.16 24.24 0.00 8 
8 1.33 13.51 0.00 5 
9 1.04 32.14 1.00 5 
10 1.33 17.14 0.00 3 
11 0.94 25.00 0.00 6 
12 1.33 12.12 0.00 4 
13 1.04 24.00 0.00 6 
14 1.28 20.00 0.00 3 
15 0.92 14.29 0.00 3 
16 1.31 20.69 0.00 3 
17 0.89 15.79 0.00 3 
18 1.06 13.64 0.00 3 
19 0.84 17.65 0.00 3 
20 1.13 13.69 0.00 3 
21 0.95 0.00 - - 
There are some results that deserve to be outlined . 
%Rec. remains rather constant in correspondence of the different shiftZ −  values with some fluctuations 
taking minima values mainly at shiftZ − = 11, 15, 19,21. 
A graph is given in Fig.39. 
Fig. 39 
Atomic Weight (%Recurrences)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
%Det. assumes rather  low values also with a length Line L=3. It oscillates among maxima and minima 
for increasing values of shiftZ − as it is pictured in Fig.40 (a, b, c). Significantly, %Det. goes definitively 
to zero starting with shiftZ − =21. 
Fig. 40a 
Atomic Weight (%Determinism)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
Rather interesting appear also the value we obtain for Entropy and Max Line as reported in the following 
figures. 
Fig. 40b 
Atomic Weight (Entropy)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
Fig. 40c 
Atomic Weight (Max-Line)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
We may now pass to consider  Recurrence Quantification Analysis in the case of Mass Numbers, )(ZA . 
The results for %Rec., %Det., ENT., and Max Line are given in Table2. 
Table 2 
Z-shift %Rec. %Det. Entropy Max-Line 
1 1.20 77.50 1.37 14 
2 1.33 30.23 0.92 7 
3 1.23 41.03 1.00 13 
4 1.43 36.36 0.81 7 
5 1.23 37.84 1.00 11 
6 1.30 21.05 1.00 5 
7 1.09 38.71 1.00 8 
8 1.44 20.00 1.00 5 
9 1.19 31.25 1.00 6 
10 1.41 8.11 0.00 3 
11 1.25 18.75 0.00 6 
12 1.33 21.21 1.00 4 
13 1.28 41.94 1.59 6 
14 1.58 27.03 0.92 4 
15 1.14 46.15 1.59 6 
16 1.45 9.38 0.00 3 
17 1.31 21.43 0.00 3 
18 1.59 21.21 1.00 4 
19 1.19 25.00 0.00 3 
20 0.92 16.67 0.00 3 
21 0.79 0.00 - - 
%Rec. remains rather constant in correspondence of the different shiftZ −  values with some fluctuations  
taking minima values mainly at shiftZ − = 7,9,11,13,15,..,19. 
A graph is given in Fig.41 (a, b, c, d) 
Fig.41a 
Mass Number (%Recurrences)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
Fig. 41b 
Mass Number (%Determinism)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
Fig. 41c 
Mass Number (Entropy)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
Fig. 41d 
Mass Number (Max-Line)
0 2 4 6 8 10 12 14 16 18 20 22 Z-shift
%Det. assumes rather  low values also with a length Line L=3. It oscillates among maxima and minima 
for increasing values of shiftZ − as it is pictured in Fig.41b. Significantly, %Det. goes definitively to zero 
starting with shiftZ − =21. 
In Fig.42 we have the comparison of %Det of Atomic Weights respect to %Det of Mass Numbers. 
     
Fig.42    
Atomic Weights - Mass Number (% Determinism)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Z, atomic number
atomic weigths
mass number
 Looking at the results given in Tables 1 and 2 and linked figures, we deduce that for different values of 
shiftZ − , the corresponding values of %Rec tend to show fluctuations. As it is well known, %Rec 
indicates in some manner presence of pseudoperiodicities. Therefore the rather small fluctuations of 
%Rec  indicate that we are in presence of a mechanism of increasing mass that tends to preserve some 
kind of periodicity and self-resemblance with rather modest fluctuations The more interesting datum is 
given by %Det. In this case we have more marked oscillations showing that in the process of increasing 
mass of stable atomic nuclei we have phase of increasing stability as opposed to phases of decrease 
stability. Here the law is the mechanism of addition of nucleons that is realized at each step between the 
given nucleus and its subsequent as considered in our phase space representation. %Det oscillations 
indicate that the process of progressively addition of nucleons in nuclei happens on the basis of a complex 
non linear mechanism in which the determinism and thus the same predictability of subsequent Mass 
Number and /or Atomic Weights is very complex and so distant from a simple and linear regime of 
addition of matter that we have expected to hold for a very long time. Looking at the values of Entropy, 
expressed in bits, one finds that also in this case oscillations are dominant for increasing values of Z-shift. 
The same happens for MaxLine whose inverse gives estimation of the divergence of the system in 
consideration giving direct indication of a possible chaotic regime. 
In conclusion, by using RQA we conclude that the mechanism of increasing mass in atomic nuclei is 
rather periodical and self-resemblance. We have obtained marked oscillations for the values of RQA 
variables. The important thing to remember here is that we are operating in a reconstructed phase space 
that takes into account o more an isolated nucleus as in the classical nuclear physics discussions, but each 
time pairs of nuclei in the embedded space with dimension D=2.The deriving behaviour of the 
mechanism of increasing mass of atomic nuclei evidences in this case all its complexity. We have now set 
of nuclei that evidence their oscillatory behaviour for % Rec, %Det, Entropy and Max Line.Such 
oscillatory behaviours of classes of nuclei result obviously connected to “periodicities” and mainly to 
classes of similarities that also stable nuclei seem to exhibit. The marked variations in the values of 
determinism indicate that the whole process results rather complex  and it is regulated from phases of 
more stability and subsequent phases of increased instability. 
In order to conclude such kind of research, and to confirm the new results that we have here indicated, we 
have  performed the last kind analysis. In this last case we have in some manner overturned the scheme of 
the previous analysis in the sense that we have selected an embedding dimension D=1. The reader will 
remember that results by FFN gave same uncertainty in selecting the values D=1 or D=2. Our previous 
RQA was performed by using D=2 . In this final exploration we use D=1. In this condition of analysis a 
given value of delay and thus , in our case of shiftZ −  , has no more sense . Each point in phase space is 
given by a value of )(ZWa or of )(ZA . To use RQA we have to select a distance , that is to say a Radius R. 
Using Euclidean distance , R will result to be the difference A∆  between two values of Mass Numbers in 
the case one utilizes )(ZA  for the analysis. In conclusion we have 
NZA += , 222 NZA += . 
The distance , R, to use in RQA will result to be given 
NZA ∆+∆=∆  
We decided to use RQA considering L=3 as Line Length and R increasing step by step from 1 to209. In 
this manner we calculated %Rec, %Det, Entropy and Max Line, for increasing values of ,.....3,2,1=Z  . 
Note that in such kind of analysis we used also shuffled data in order to ascertain the validity of the 
obtained results. In addition , on the obtained data, we used also a Wald-Wolfowitz run test that we 
executed on %Det and on %Det / %Rec, and  the probability that the results were obtained by chance, 
was found to be <0.001. 
The results are now given in Tables 3, 4 and in Figures 43, 44, 45, 46, 47. 
Table 3: Results of Recurrences quantification analysis of Mass Number with embedding D=1 
  and distance R ranging from 1 to 209 
distance 
Rec. 
Det. %Det./%Rec. 
distance 
Rec. 
Det. %Det./%Rec.
distance
Rec. 
Det. %Det./%Rec. 
1 0.029 0.000 0.000 71 42.051 98.742 2.348 141 71.907 99.55 1.384 
2 1.029 8.571 8.329 72 42.051 98.742 2.348 142 72.436 99.513 1.374 
3 1.381 23.404 16.947 73 42.786 98.283 2.297 143 72.436 99.513 1.374 
4 1.381 23.404 16.947 74 43.432 98.241 2.262 144 72.877 99.395 1.364 
5 2.204 66.667 30.248 75 43.432 98.241 2.262 145 73.288 99.399 1.356 
6 3.115 75.472 24.229 76 44.167 99.069 2.243 146 73.288 99.399 1.356 
7 4.29 82.192 19.159 77 44.902 98.822 2.201 147 73.817 99.602 1.349 
8 4.29 82.192 19.159 78 45.519 98.773 2.170 148 74.376 99.526 1.338 
9 5.172 86.364 16.698 79 45.519 98.773 2.170 149 74.904 99.451 1.328 
10 6.024 89.268 14.819 80 46.195 98.885 2.141 150 74.904 99.451 1.328 
11 6.024 89.269 14.819 81 46.812 98.87 2.112 151 75.316 99.532 1.322 
12 7.082 90.45 12.772 82 46.812 98.87 2.112 152 75.727 99.728 1.317 
13 7.875 88.806 11.277 83 47.458 98.885 2.084 153 75.725 99.728 1.317 
14 8.639 91.156 10.552 84 48.193 98.963 2.053 154 76.197 99.691 1.308 
15 8.639 91.156 10.552 85 48.78 98.675 2.023 155 76.609 99.501 1.299 
16 9.58 94.785 9.894 86 48.78 98.675 2.023 156 77.05 99.657 1.293 
17 10.638 94.475 8.881 87 49.398 98.989 2.004 157 77.05 99.657 1.293 
18 10.638 94.475 8.881 88 50.132 99.062 1.976 158 77.667 99.659 1.283 
19 11.666 93.703 8.032 89 50.132 99.062 1.976 159 78.078 99.511 1.275 
20 12.401 95.261 7.682 90 50.779 99.016 1.950 160 78.078 99.511 1.275 
21 13.282 95.575 7.196 91 51.455 99.029 1.925 161 78.343 99.4 1.269 
22 13.282 95.575 7.196 92 52.16 99.155 1.901 162 78.754 99.664 1.266 
23 14.252 94.433 6.626 93 52.16 99.155 1.901 163 79.224 99.703 1.258 
24 15.046 95.508 6.348 94 52.806 98.998 1.875 164 79.224 99.703 1.258 
25 15.046 95.508 6.348 95 53.425 99.065 1.854 165 79.783 99.595 1.248 
26 16.927 97.232 5.744 96 53.425 99.065 1.854 166 80.194 99.45 1.240 
27 18.513 96.508 5.213 97 54.041 99.402 1.839 167 80.635 99.745 1.237 
28 17.602 95.993 5.454 98 54.687 99.087 1.812 168 80.635 99.745 1.237 
29 17.602 95.993 5.454 99 55.245 98.83 1.789 169 81.105 99.565 1.228 
30 18.513 96.508 5.213 100 55.245 98.83 1.789 170 81.399 99.458 1.222 
31 19.16 96.626 5.043 101 55.804 99.052 1.775 171 81.399 99.458 1.222 
32 19.16 96.625 5.043 102 56.303 98.904 1.757 172 81.781 99.748 1.220 
33 19.982 96.176 4.813 103 56.92 99.019 1.740 173 82.222 99.607 1.211 
34 21.04 97.067 4.613 104 56.92 99.019 1.740 174 82.662 99.523 1.204 
35 21.863 97.312 4.451 105 57.743 99.237 1.719 175 82.662 99.573 1.205 
36 21.863 97.312 4.451 106 58.419 98.994 1.695 176 83.133 99.611 1.198 
37 22.656 97.017 4.282 107 58.419 98.994 1.695 177 83.368 99.612 1.195 
38 23.45 97.243 4.147 108 58.919 99.202 1.684 178 83.368 99.612 1.195 
39 24.42 96.51 3.952 109 59.536 99.112 1.665 179 83.75 99.684 1.190 
40 24.42 96.51 3.952 110 60.182 99.121 1.647 180 84.337 99.617 1.181 
41 25.272 96.86 3.833 111 60.182 99.121 1.647 181 84.69 99.722 1.177 
42 25.918 96.485 3.723 112 60.711 99.177 1.634 182 84.69 99.722 1.177 
43 25.918 96.485 3.723 113 61.387 99.234 1.617 183 84.984 99.654 1.173 
44 26.8 96.82 3.613 114 61.387 99.234 1.617 184 85.307 99.724 1.169 
45 27.593 97.551 3.535 115 62.033 99.337 1.601 185 85.307 99.724 1.169 
46 28.299 98.027 3.464 116 62.504 99.436 1.591 186 85.688 99.863 1.165 
47 28.299 98.027 3.464 117 63.033 99.441 1.578 187 86.13 99.693 1.157 
48 29.151 97.984 3.361 118 63.033 99.441 1.578 188 86.483 99.728 1.153 
49 29.944 97.544 3.258 119 63.562 99.353 1.563 189 86.483 99.728 1.153 
50 29.944 97.547 3.258 120 64.091 99.358 1.550 190 86.835 99.763 1.149 
51 30.767 98.376 3.197 121 64.091 99.358 1.550 191 87.129 99.865 1.146 
52 31.472 98.039 3.115 122 64.678 99.5 1.538 192 87.129 99.865 1.146 
53 32.119 98.079 3.054 123 65.207 99.459 1.525 193 87.423 99.832 1.142 
54 32.119 98.079 3.054 124 65.795 99.285 1.509 194 87.775 99.766 1.137 
55 30.059 98.489 3.277 125 65.795 99.285 1.509 195 88.099 99.666 1.131 
56 33.911 98.44 2.903 126 66.441 99.513 1.498 196 88.099 99.666 1.131 
57 33.911 98.44 2.903 127 66.941 99.517 1.487 197 88.481 99.801 1.128 
58 34.558 98.469 2.849 128 66.941 99.517 1.487 198 88.804 99.735 1.123 
59 35.292 98.751 2.798 129 67.382 99.477 1.476 199 89.039 99.736 1.120 
60 36.086 98.616 2.733 130 67.852 99.524 1.467 200 89.039 99.736 1.120 
61 36.086 98.616 2.733 131 68.44 99.614 1.455 201 89.362 99.704 1.116 
62 36.938 98.329 2.662 132 68.44 99.614 1.455 202 89.686 99.803 1.113 
63 37.702 98.051 2.601 133 69.486 99.49 1.432 203 89.686 99.803 1.113 
64 37.702 98.051 2.601 134 69.556 99.493 1.430 204 90.009 99.739 1.108 
65 38.29 98.388 2.570 135 69.909 99.496 1.423 205 90.332 99.707 1.104 
66 39.024 98.494 2.524 136 69.906 99.496 1.423 206 90.538 99.838 1.103 
67 39.788 98.523 2.476 137 70.291 99.373 1.414 207 90.538 99.838 1.103 
68 39.788 98.523 2.476 138 70.79 99.377 1.404 208 90.831 99.871 1.100 
69 40.406 98.545 2.439 139 70.79 99.377 1.404 209 91.155 99.742 1.094 
70 41.17 98.787 2.399 140 71.378 99.506 1.394         
Fig. 43 
Fig. 44 
Fig. 45 
Fig. 46 
Fig. 47 
The obtained results may be considered of valuable interest since they indicate possible new properties 
for Mass Number of atomic nuclei. 
At increasing values of Radius R, % Rec and % Det increase, as it is trivially expected in some general 
case, but the interesting new thing is that, after some regular increasing values of %Rec and %Det, 
occurring every two or three step, soon after the values of RQA variables reach values of stability that so 
remain for two steps in the increasing values of R. In other terms,  in presence of increasing R, we have  
corresponding increasing values of % Rec, %Det, Entropy, followed by a phase in which, still for 
increasing R, the values of RQA variables remain instead constant.  
This is certainly a new mechanism of increasing mass of atomic nuclei that deserves to be carefully 
explained. 
Table 4 
d=2 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=3 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=4 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N    
  3 4 4 5 1 1     1 0 2 2 1 2     3 4 5 6 2 2 
  4 5 5 6 1 1     2 2 3 4 1 2     6 6 8 8 2 2 
  6 6 7 7 1 1     4 5 6 6 2 1     8 8 10 10 2 2 
  7 7 8 8 1 1     5 6 7 7 2 1     9 10 11 12 2 2 
  26 30 28 30 2 0     8 8 9 10 1 2     10 10 12 12 2 2 
  38 50 40 50 2 0     10 10 11 12 1 2     11 12 13 14 2 2 
  52 78 54 78 2 0     12 12 13 14 1 2     12 12 14 14 2 2 
  56 82 58 82 2 0     14 14 15 16 1 2     13 14 15 16 2 2 
  57 82 59 82 2 0     16 16 17 18 1 2     14 14 16 16 2 2 
  66 98 68 98 2 0     21 24 22 26 1 2     15 16 17 18 2 2 
  77 116 78 117 1 1     22 26 23 28 1 2     17 18 19 20 2 2 
  78 117 79 118 1 1     24 28 25 30 1 2     22 26 24 28 2 2 
                  25 30 28 30 3 0     23 28 25 30 2 2 
                  26 30 27 32 1 2     24 28 26 30 2 2 
                  37 48 38 50 1 2     25 30 27 32 2 2 
                  40 50 41 52 1 2     27 32 29 34 2 2 
                  43 56 44 58 1 2     33 42 35 44 2 2 
                  45 58 46 60 1 2     34 46 36 48 2 2 
                  52 78 55 78 3 0     36 48 38 50 2 2 
                  56 82 59 82 3 0     37 48 39 50 2 2 
                  59 82 60 84 1 2     39 50 41 52 2 2 
                  68 98 69 100 1 2     42 56 44 58 2 2 
                  73 108 74 110 1 2     43 56 45 58 2 2 
                  74 110 75 112 1 2     44 58 46 60 2 2 
                  76 116 78 117 2 1     45 58 47 60 2 2 
                  80 122 81 124 1 2     58 82 60 84 2 2 
                  81 124 82 126 1 2     59 82 61 84 2 2 
                                  67 98 69 100 2 2 
                                  72 108 74 110 2 2 
                                  77 116 79 118 2 2 
                                  81 124 83 126 2 2 
                                              
d=6 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=7 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=8 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N    
  1 0 3 4 2 4     2 2 5 6 3 4     1 0 4 5 3 5 
  7 7 10 10 3 3     3 4 7 7 4 3     2 2 6 6 4 4 
  19 20 21 24 2 4     4 5 8 8 4 3     5 6 9 10 4 4 
  21 24 23 28 2 4     6 6 9 10 3 4     6 6 10 10 4 4 
  24 28 28 30 4 2     8 8 11 12 3 4     8 8 12 12 4 4 
  28 30 30 34 2 4     10 10 13 14 3 4     9 10 13 14 4 4 
  29 34 31 38 2 4     12 12 15 16 3 4     10 10 14 14 4 4 
  31 38 33 42 2 4     14 14 17 18 3 4     11 12 15 16 4 4 
  32 42 34 46 2 4     16 16 19 20 3 4     12 12 16 16 4 4 
  35 44 37 48 2 4     21 24 24 28 3 4     13 14 17 18 4 4 
  36 48 40 50 4 2     22 26 25 30 3 4     15 16 19 20 4 4 
  41 52 43 56 2 4     23 28 28 30 5 2     16 16 18 22 2 6 
  48 66 50 70 2 4     24 28 27 32 3 4     16 16 20 20 4 4 
  49 66 51 70 2 4     26 30 29 34 3 4     18 22 22 26 4 4 
  51 70 53 74 2 4     43 56 46 60 3 4     20 20 22 26 2 6 
  53 74 55 78 2 4     47 60 48 66 1 6     22 26 26 30 4 4 
  54 78 56 82 2 4     48 66 51 70 3 4     23 28 27 32 4 4 
  55 78 57 82 2 4     50 70 53 74 3 4     25 30 29 34 4 4 
  56 82 60 84 4 2     54 78 57 82 3 4     26 30 30 34 4 4 
  57 82 61 84 4 2     55 78 58 82 3 4     34 46 38 50 4 4 
  62 90 64 94 2 4     56 82 61 84 5 2     37 48 41 52 4 4 
  63 90 65 94 2 4     61 84 62 90 1 6     40 50 42 56 2 6 
  64 94 66 98 2 4     62 90 65 94 3 4     42 56 46 60 4 4 
  65 94 67 98 2 4     64 94 67 98 3 4     43 56 47 60 4 4 
  69 100 71 104 2 4     65 94 68 98 3 4     46 60 48 66 2 6 
  70 104 72 108 2 4     70 104 73 108 3 4     47 60 49 66 2 6 
  71 104 73 108 2 4     72 108 75 112 3 4     52 78 56 82 4 4 
  73 108 75 112 2 4     78 117 80 122 2 5     54 78 58 82 4 4 
  75 112 77 116 2 4     80 122 83 126 3 4     55 78 59 82 4 4 
  80 122 82 126 2 4                     60 84 62 90 2 6 
                                  61 84 63 90 2 6 
                                  64 94 68 98 4 4 
                                  68 98 70 104 2 6 
                                  74 110 76 116 2 6 
                                  75 112 78 117 3 5 
                                  79 118 81 124 2 6 
                                              
d=9 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=10 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=11 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N    
  3 4 8 8 5 4     1 0 5 6 4 6     1 0 6 6 5 6 
  5 6 10 10 5 4     2 2 7 7 5 5     4 5 10 10 6 5 
  7 7 11 12 4 5     4 5 9 10 5 5     6 6 11 12 5 6 
  9 10 14 14 5 4     7 7 12 12 5 5     8 8 13 14 5 6 
  11 12 16 16 5 4     17 18 21 24 4 6     10 10 15 16 5 6 
  15 16 18 22 3 6     21 24 25 30 4 6     12 12 17 18 5 6 
  15 16 20 20 5 4     22 26 28 30 6 4     14 14 19 20 5 6 
  19 20 22 26 3 6     27 32 31 38 4 6     18 22 23 28 5 6 
  25 30 30 34 5 4     30 34 32 42 2 8     20 20 23 28 3 8 
  33 42 36 48 3 6     31 38 35 44 4 6     21 24 26 30 5 6 
  34 46 39 50 5 4     32 42 36 48 4 6     22 26 27 32 5 6 
  35 44 38 50 3 6     33 42 37 48 4 6     24 28 29 34 5 6 
  36 48 41 52 5 4     34 46 40 50 6 4     28 30 31 38 3 8 
  39 50 42 56 3 6     35 44 39 50 4 6     29 34 32 42 3 8 
  40 50 43 56 3 6     38 50 42 56 4 6     30 34 33 42 3 8 
  41 52 44 58 3 6     39 50 43 56 4 6     31 38 34 46 3 8 
  42 56 47 60 5 4     41 52 45 58 4 6     32 42 37 48 5 6 
  46 60 49 66 3 6     50 70 52 78 2 8     35 44 40 50 5 6 
  51 70 52 78 1 8     52 78 58 82 6 4     38 50 43 56 5 6 
  52 78 57 82 5 4     65 94 69 100 4 6     45 58 48 66 3 8 
  54 78 59 82 5 4     66 98 70 104 4 6     51 70 54 78 3 8 
  60 84 63 90 3 6     67 98 71 104 4 6     52 78 59 82 7 4 
  67 98 70 104 3 6     70 104 74 110 4 6     53 74 56 82 3 8 
  68 98 71 104 3 6     75 112 79 118 4 6     55 78 60 84 5 6 
  71 104 74 110 3 6     76 116 80 122 4 6     59 82 62 90 3 8 
  74 110 77 116 3 6     78 117 81 124 3 7     63 90 66 98 3 8 
  77 116 80 122 3 6                     64 94 69 100 5 6 
                                  66 98 71 104 5 6 
                                  69 100 72 108 3 8 
                                  73 108 76 116 3 8 
                                  74 110 78 117 4 7 
                                  79 118 82 126 3 8 
                                              
d=13 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=14 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N      d=15 Z N Z N ∆∆∆∆ Z    ∆∆∆∆ N    
  1 0 7 7 6 7     4 5 11 12 7 7     1 0 8 8 7 8 
  3 4 10 10 7 6     7 7 14 14 7 7     2 2 9 10 7 8 
  5 6 12 12 7 6     15 16 21 24 6 8     4 5 12 12 8 7 
  7 7 13 14 6 7     21 24 27 32 6 8     6 6 13 14 7 8 
  9 10 16 16 7 6     25 30 31 38 6 8     8 8 15 16 7 8 
  13 14 18 22 5 8     32 42 38 50 6 8     10 10 17 18 7 8 
  13 14 20 20 7 6     33 42 39 50 6 8     12 12 19 20 7 8 
  16 16 21 24 5 8     35 44 41 52 6 8     18 22 25 30 7 8 
  17 18 22 26 5 8     36 48 42 56 6 8     20 20 25 30 5 10 
  19 20 24 28 5 8     37 48 43 56 6 8     22 26 29 34 7 8 
  21 24 28 30 7 6     38 50 44 58 6 8     27 32 32 42 5 10 
  23 28 30 34 7 6     39 50 45 58 6 8     30 34 35 44 5 10 
  26 30 31 38 5 8     41 52 47 60 6 8     31 38 36 48 5 10 
  33 42 38 50 5 8     46 60 50 70 4 10     32 42 39 50 7 8 
  34 46 41 52 7 6     47 60 51 70 4 10     33 42 40 50 7 8 
  37 48 42 56 5 8     52 78 60 84 8 6     36 48 43 56 7 8 
  39 50 44 58 5 8     53 74 59 82 6 8     38 50 45 58 7 8 
  40 50 45 58 5 8     56 82 62 90 6 8     43 56 48 66 5 10 
  41 52 46 60 5 8     57 82 63 90 6 8     46 60 51 70 5 10 
  44 58 49 66 5 8     60 84 64 94 4 10     49 66 52 78 3 12 
  47 60 50 70 3 10     61 84 65 94 4 10     52 78 61 84 9 6 
  48 66 53 74 5 8     62 90 68 98 6 8     56 82 63 90 7 8 
  50 70 55 78 5 8     68 98 72 108 4 10     60 84 65 94 5 10 
  53 74 58 82 5 8     73 108 78 117 5 9     65 94 70 104 5 10 
  54 78 61 84 7 6     78 117 83 126 5 9     67 98 72 108 5 10 
  57 82 62 90 5 8                     68 98 73 108 5 10 
  58 82 63 90 5 8                     69 100 74 110 5 10 
  61 84 64 94 3 10                     72 108 78 117 6 9 
  62 90 67 98 5 8                     75 112 80 122 5 10 
  63 90 68 98 5 8                     77 116 82 126 5 10 
  70 104 75 112 5 8                                 
  72 108 77 116 5 8                                 
  74 110 79 118 5 8                                 
  76 116 81 124 5 8                                 
  78 117 82 126 4 9                                 
In Table 4 we give the scheme of increasing R corresponding to A∆  and the corresponding variations in 
the number of nucleons as they are induced step by step. Obviously this table 4 cannot be complete. 
However,  the exposition of the process, also limited to few cases of interest, will contribute to elucidate 
the mechanism under consideration. In brief for 2=∆A  we have oscillation in the values of RQA 
variables but they soon after return to be stable for 3=∆A and 4=∆A . After we pass to 6=∆A where again 
RQA variables are unstable but they return to be stable for 7=∆A and 8=∆A . The next step is 9=∆A with 
instability, followed from stable values for 10=∆A and 11=∆A . We may continue with 13=∆A  that is 
unstable but followed from stable 14=∆A and 15=∆A .The same thing happens for 
...........180.............119........80........38343027232016 ororororororororororororA =∆ . To each given 
unstable A∆  value , will correspond two subsequent stable values that respectively will be given at 
17=∆A and 18=∆A ; at 21=∆A and 22=∆A , ……….. at 120=∆A and 121=∆A , ….. at 181=∆A and 
182=∆A . 
Instabilities are present every three or four increasing values of A∆ . Systematically, each of them is 
followed by stable values at the two subsequent increments of A∆ . 
In conclusion the law seems as it follows: for each pair of nuclei , fixed the value of A∆  with unstable 
value of the RQA variables, the addition of one nucleon by two subsequent steps   stabilizes the values of 
such variables. Obviously, for each selected value of A∆  we have a class of pair of nuclei as indicated as 
example in Table 4. 
In conclusion, the use of RQA variables has cleared that we are in presence of new features for atomic 
nuclei that deserve to be properly explained. We intend to say that the next step of the present research 
should be now to link the different results that have been obtained with  concrete evidences expressible in 
terms of basic concepts of nuclear physics. If on one hand some of such new findings are just evident by 
itself  on the other hand we cannot ignore that in this paper we have moved more on the line of the 
notions as they are contained in the methods that we have used. More concretely : referring as example to 
the basic results that we have obtained by using RQA, and, in particular, to the last results as given by 
using embedding dimension D=1 and reported in Tables 3 and 4 and in Figures 43-47, we cannot ignore 
that we have to consider now pairs of nuclei with given A∆  and thus to identify pairs of subsequent stable 
nuclei and, following this way, to find some new regularities in Z, N and to give new classifications of 
nuclei to different groups using such regularities. In short, the results that we have obtained should reveal 
new regularities about ground states of nuclei not found so easily by other methods. Consequently, this 
new approach might be very useful and important. The aim is to pursue such research work in our future 
investigations. 
8. Conclusions 
In the present paper we have introduced a preliminary but complete analysis of Atomic Weights and Mass 
Number using the methods of non linear analysis. 
We have obtained some  results that appear to be of some interest in understanding the basic foundations 
of nuclear matter. As methodology, we have applied the tests of autocorrelation function and of Mutual 
Information. We have also provided to a reconstruction of the experimental data in phase space giving 
results on Lyapunov spectrum and Correlation Dimension. We have performed an analysis to establish 
the presence of a power law in data on Atomic Weights and Mass Number and such kind of analysis has 
been completed by using the technique of the variogram. The results seem to confirm the presence of a 
fractal regime in the process of increasing mass of atomic nuclei. The estimation of Husrt exponent has 
enabled us to indicate that we would be in presence of a fractional Brownian regime with long range 
correlations.  
To summarize: Some preliminary results have been obtained. The mechanism of increasing mass in 
atomic nuclei reveals itself to be  a nonlinear mechanism marked by a non integer value of Correlation 
dimension in phase space reconstruction. The presence of positive Lyapunov exponents indicate that the 
system of mass increasing is divergent and thus possibly chaotic. By using an identified Power Law and 
the variogram technique we may conclude that we are in presence of a fractal regime, a fractional 
Brownian regime. 
The most relevant results have been  obtained by using RQA. The process under our investigation results 
to be not fully deterministic when considering an embedding dimension D=2. We are in presence of self-
resemblance and pseudo periodicities that show small fluctuations at increasing value of shiftZ −  while 
instead Determinism shows consistent variations at increasing values of such parameter. Also Entropy 
and Max Line reveal the same tendency. Therefore, in the same framework of stable nuclei we have phase 
of increasing stability or increasing instability, depending on the mechanism of composition of the 
considered atomic nuclei and on the differences that they exhibit in the values of their Atomic Weights 
and of Mass Number. A final important result is obtained by using RQA in phase space reconstruction 
using embedding dimension D=1 and increasing Radius R corresponding to net differences in Mass 
number of the considered atomic nuclei. In this case, in phase space reconstruction, RQA involves pairs 
of nuclei in our analysis. New properties are identified at the increasing values of A∆ . In particular, 
determinism oscillates but at some regular distances it also shows definite constant values as well as the 
other RQA variables . This confirms that we are in presence of a mechanism of increasing mass of atomic 
nuclei in which phases of stability result subsequent to phases of instability possibly marked from 
conditions of order-disorder like transitions. We have to consider pairs of nuclei with fixed A∆  and to 
identify pairs of subsequent stable nuclei that indicate new regularities in Z, N that we need to indicate in 
detail . We have to classify nuclei pertaining to different groups using these new regularities. This 
approach might be of valuable interest and it will constitute the object of our future work.  
In this framework, the next step of the present investigation will be also  to analyze data corresponding to 
values of binding energies for atomic nuclei. Possibly the complex of such results will give the possibility 
to indicate new perspectives in the elaboration of more accurate nuclear models of nuclear matter. 
Acknowledgement 
Many thanks are due to M. Pitkanen for his continuous and stimulating interest, suggestions and 
encouragement through this work. 
Software NDT by J. Reiss and VRA by E. Kononov were also used for general non linear analysis. 
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