# Boiling Flow Pattern Identification Using a Self-Organizing Map
## 2 Materials and Methods
### Recurrence Quantification Analysis  
In order to detect changes in the dynamics of pressure drop oscillations, a recurrence window analysis was used. This method is based on a quantitative analysis of recurrence plots (_RP_) [23]. [PERSON] et al. suggested a two-dimensional representation of the recurrences of a dynamical system according to the formula [24]:  
\[R_{i,j}=\theta\Big{(}\varepsilon_{i}-\|x_{i}-x_{j}\|\Big{)},\ x_{i}\in R^{m}, \ i,j=1\dots N \tag{1}\]  
where \(N\) is the number of considered states \(x\), \(\varepsilon\) is a threshold distance, \(\|\,\|\) is a norm in \(m\)-dimensional space, \(\theta\) is the Heaviside function, and \(R\) is a set of real numbers. In order to compute an _RP_, appropriate parameters for attractor reconstruction must be estimated [25]. In this paper, the appropriate time delay \(\tau\) was estimated as a \(\tau\) for which the mutual information [26] vs. time delay reaches its first minimum. The embedding dimension \(m\) was obtained by the algorithm of a fraction of false nearest neighbors. For pressure fluctuations in the  
Figure 1: Schematic diagram of the experimental setup: (**1**) supply tank; (**2**) ball valve; (**3**) surge tank; (**4**) heated minichannel; (**5**) pressure drop sensor; (**6**) high-speed camera; (**7**) overflow tank. Figure 2: Maximum, minimum, and average values of the pressure drop oscillations during a single pressure drop oscillation (_PDO_) cycle for \(P\) = 46 W and \(G\) = 159 kg/m\({}^{2}\)s.  
period of time between 101-106 s (Figure 2), the first minimum was reached for \(\tau\) equal to 100 samples (0.1 s). The false nearest neighbor algorithm estimated \(m\) as equal being to 4. The results of the estimation of \(\tau\) and \(m\) are shown in Figure 3.  
The important parameter of the \(RP\) is the threshold \(\varepsilon\). If the value of \(\varepsilon\) is too small, the \(RP\) would have a small number of recurrence points. On the other hand, if \(\varepsilon\) was too big, \(RP\) becomes black [23]. Several methods for selecting \(\varepsilon\) have been recommended in the literature [28; 29; 30; 31]. In this paper, it was assumed that the value of \(\varepsilon\) was determined by the size of the reconstructed attractor in the \(m\)-dimensional phase space and was 10% of the maximum attractor diameter. The example of \(RP\) for pressure fluctuations recorded in the period of time between 101-106 s is shown in Figure 4.  
The visible points' arrangement on the \(RP\) characterizes the properties of the dynamic system [23]. In order to achieve their quantitative determining, [PERSON] and [PERSON] [30] proposed a method called Recurrence Quantification Analysis (\(RQA\)). In this paper, the four \(RQA\) coefficients are used to describe the characteristic states occurring in the \(RP\) (single points, diagonal line, vertical line, horizontal  
Figure 4: Recurrence plot of pressure fluctuations in the period of time between 101–106 s for \(m=4\), \(\tau=100\), \(\varepsilon=0.8\sigma\). The calculations have been made using the CRP toolbox for Matlab®[27]. Figure 3: Estimation of \(\tau\) and \(m\) for pressure fluctuations in the period of time between 101–106 s (Figure 2): (**a**) false nearest neighbors as a function of the embedding dimension; (**b**) mutual information as a function of time delay. The calculations have been made using the CRP toolbox for Matlab®[27]. line). The following \(RQA\) coefficients, the values of which change significantly in a single \(PDO\) cycle, are considered:  
The determinism, \(DET\), is a percentage measure of recurrence points that belong to diagonal lines on \(RP\). Determinism is defined as follows [23]:  
\[DET=\frac{\sum_{l=l_{\min}}^{N}{IP(l)}}{\sum_{l=l}^{N}{IP(l)}} \tag{2}\]  
where \(P(l)\) defines the distribution of diagonal line lengths \(l\), and \(N\) is the length of the time series. Diagonal lines in \(RP\) are created by the parallel segments of attractor trajectories.