## 2 Materials and Methods
### DMA-Algorithm for the Identification of Periods in Data Arrays  
Within the framework of this study, the authors have developed a mathematical algorithm to identify periods (stable cycles) in data arrays. The algorithm is based on the DMA concept. DMA was briefly discussed above. The developed algorithm makes it possible to evaluate the consistency of any positive number as the period of the original function. The best answers, if they exist, are declared the identified periods. For simplicity of presentation, below we will call the algorithm developed here the DMA-algorithm. Let's move on to a rigorous mathematical description of the DMA-algorithm.  
Suppose that a function \(f\) is given on the segment \([a,b]\), and we want to understand what periods it has. The period \(T\), ideal for \(f\), is defined as follows: \(T\) is the period for \(f\) if \(\forall t\), \(t+T\in[a,b]\) is true \(f(t)=f(t+T)\). In reality, the function \(f\) may not have any ideal period \(T\), but it may have approximate periods. A description of approximate periods and how to determine them is provided below.  
It is clear that the applicent for the period \(T\) must lie in the interval \((0,\,(b-a)/2)\). For such a \(T\) and \(t\in[a,\,a+T)\), we denote by \(\{f,\,T,\,t\}\) the finite sequence \(\{f(t),f(t+T),\,f(t+2T),\,\dots\,\}\). Firstly, it is necessary to understand to what extent the sequence \(\{f,\,T,\,t\}\) can be considered constant, i.e., to determine the exponent of its constancy \(C[f,\,T,\,t]\geq 0\). The equality \(C[f,\,T,\,t]=0\) must correspond to the constancy of the sequence \(\{f,\,T,\,t\}\).  
Here we present three different constructions of the exponent \(C[f,\,T,\,t]\).  
The first construction is the variance \(D\) of the sequence \(\{f,\,T,\,t\}\) relative to the usual or average median:  
\[C[f,\,T,\,t]=D[f,\,T,\,t]\text{ or }C[f,\,T,\,t]=D_{m}\{f,\,T,\,t\}. \tag{1}\]  
The second and third constructions suppose the determination of the modulus of the difference sequence \(\,\mid\Delta[f,\,T,\,t]\mid\,\) for sequence \(\{f,\,T,\,t\}\):  
\[\mid\Delta[f,\,T,\,t]\mid\,=\{\,\mid f(t+T)-f(t)\,\mid,\,f(t+2T)-f(t+T)\,\mid,\dots\,\}. \tag{2}\]  
The question of the constancy of \(\{f,\,T,\,t\}\) is reduced to the question of the triviality of \(\mid\Delta[f,\,T,\,t]\mid\,\).  
The second construction is the Kolmogorov mean \(M_{p}\) with a positive exponent \(p\) of the sequence \(\,\mid\Delta[f,\,T,\,t]\mid\,\):  
\[C[f,\,T,\,t]=M_{p}\mid\Delta[f,\,T,\,t]\mid. \tag{3}\]  
As mentioned previously, using the Kolmogorov mean \(M_{p}\), \(C\) indicates the proximity of the sequence \(\,\mid\Delta[f,\,T,\,t]\mid\,\) to zero.  
Another solution is given by the third construction, using the distribution function \(F_{\,\mid\Delta[f,\,T,\,t]\mid\,\):  
\[C[f,\,T,\,t]=\alpha(\,\mid\Delta[f,\,T,\,t]\mid\,), \tag{4}\]  
where \(\alpha(\,\mid\Delta[f,\,T,\,t]\mid\,)\) is a-quantile of the distribution \(F_{\,\mid\Delta[f,\,T,\,t]\mid\,\).  
The estimate \(C[f,\,T,\,t]\) characterizes \(T\) as a period on the sequence \(\{t,\,t+T,\,t+2T,\,\dots\,\}\).  
The next stage is the development of a unified estimate \(T\) as a period independent of \(t\). Such an estimate \(C(f,\,T)\) should be an indicator of the smallness of the set (\(C(f,\,T,\,t)\), \(t\in[a,\,a+T)\)). For this, we again use the Kolmogorov mean \(M_{p}\):  
\[C(f,\,T)=M_{p}(C(f,\,T,\,t),\,t\in[a,\,a+T)),\,p\geq 0. \tag{5}\]  
The final stage is the search for strong minima of the estimate \(C(f,\,T)\). Provided that they exist, we can obtain the necessary periods for the function \(f\) using the apparatus of minimality measures [4].