The dynamics of the underlying phenomena can be investigated by means of recurrences that are calculated for each visited state of the reconstructed trajectory. This method was developed by Eckmann [13] and extended by Webber [14], Casdagli [15] later by Marwan et al. [16, 17] and others. This approach was used for both short deterministic and noise-affected experimental data [18, 19].

Two points on a trajectory are marked as neighbours if they are close enough to each other. This can be expressed by the distance matrix *R* with its element $${R}_{ij}^{\u03f5}$$ given by [13]:

$${R}_{ij}^{\u03f5}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Theta \mathrm{(}\u03f5\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\parallel {x}_{i}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{x}_{j}\parallel \mathrm{)},\text{\hspace{1em}(1)}$$(1)

where ϵ is the threshold value and Θ(*x*) denotes the Heaviside function. The number of recurrence points depends on both the underlying dynamics and the cosing the threshold value. A standard technique for approximations is that the threshold value should not be higher than a few percentage of the total number of points [17].

According to Takens [20] and the laser light transmission *x* time series, the following vectors **x** in the embedding space are defined:

$$\begin{array}{c}x\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{(}x\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}x\mathrm{(}t\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\delta i\Delta t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}\text{\hspace{0.05em}}x\mathrm{(}t\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\delta i\Delta t\mathrm{)},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}\\ x\mathrm{(}t\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{(}m\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1\mathrm{)}\mathrm{}\delta i\Delta t\mathrm{)},\end{array}\text{\hspace{1em}(2)}$$(2)

where Δ*t*=0.1 ms is a sampling period. The phase space reconstruction is performed by using the standard methods of the first minimum of average mutual information and the nodal fraction of false neighbours [21–23]. The same embedding methods are used for comparison with the assumption of the smallest time delay *δi* and then the highest dimension *m*. In this way, the system could be described the dimension *m*=7 and the delay *δi*=7.

Figure 3a–f (in the lower panels) compare the recurrence plots (RP) of the threshold ϵ = 3*σ*, where *σ* denotes the standard deviation of a particular time series depicted in the upper panels. It is noted that one criterion is used to all the cases in order to obtain 7 % recurrences in the minimal case. To estimate the distances, the Euclidean distance in the reconstructed phase space of the laser light transmission time series for six representative cases, obtained from the flows shown in Figure 2, was used. The horizontal and vertical axes represent the time instants *i* and *j* to which the distance formula (1) is applied.

Here, some interesting observations can be made. The clear checkerboard structure, which can be seen in most of figures, implies two states of intermittency [24, 25] in the light transmission time series; however, its type can be assessed with changing *q*. Here, these intermittences are mostly related to switching between better and worse transparency depending on the presence of slugs and/or bubbles. It is worth noting that in the presence of very short slugs mediated by the small bubbles (Fig. 3f) the laser light is strongly scattered by moving surface of the slugs and bubbles resulting in fast oscillations of *x*.

Figure 3c shows a large number of short slugs rarely mediated by air bubbles. Figure 3c,e, and f illustrate the structure smearing that could correspond to the larger complexity in bubbles and slugs sizes, and, consequently, to their speeds. On the other hand, more regular points distributions are visible in Figure 3a,b, and d. These cases correspond to the observations of longer air slugs and elongated bubbles.

Webber and Zbilut [14] and later Marwan et al. [16, 17, 26] developed the recurrence quantification analysis (RQA) for recurrence plots. The RQA analysis includes the recurrence rate variable, *RR*, which expresses the system’s ability to return to the neighbourhood of previous state and has the following definition:

$$RR\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{N}^{2}}{\displaystyle \sum _{i\mathrm{,}j\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}i}^{N}}{R}_{ij}^{\u03f5}.\text{\hspace{1em}(3)}$$(3)

Furthermore, the RQA was used to identify diagonal lines through their maximal lengths, *L*_{max}. The RQA provides the probability *p*(*l*) or *p*(*v*) of line distribution according to their lengths *l* or *v* (for diagonal and vertical lines). They are calculated as follows:

$$p\mathrm{(}z\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{P}^{\u03f5}\mathrm{(}z\mathrm{)}}{{\displaystyle {\sum}_{z\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{z}_{\text{min}}}^{{N}_{z}}{P}^{\u03f5}\mathrm{(}z\mathrm{)}}},\text{\hspace{1em}(4)}$$(4)

where *z*=*l* or *v* depend on diagonal or vertical structures in a specific recurrence plot. *P*^{ϵ}(*z*) denotes the histogram of *z* lengths and a fixed value of ϵ, *N*_{z} is the number of diagonal or vertical lines (depending on the definition of *z*). Measures, such as determinism *DET*, laminarity *LAM*, trapping time *TT*, and average length *L*, are based on probabilities *P*^{ϵ}(*z*)

$$DET\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\displaystyle {\sum}_{l\text{\hspace{0.17em}}\mathrm{=}\text{\hspace{0.17em}}{l}_{\text{min}}}^{{N}_{l}}l{P}^{\u03f5}\mathrm{(}l\mathrm{)}}}{{\displaystyle {\sum}_{l\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{{N}_{l}}l{P}^{\u03f5}\mathrm{(}l\mathrm{)}}},\text{\hspace{1em}(5)}$$(5)

$$LAM\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\displaystyle {\sum}_{v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{v}_{\text{min}}}^{{N}_{v}}v{P}^{\u03f5}\mathrm{(}v\mathrm{)}}}{{\displaystyle {\sum}_{v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{{N}_{v}}v{P}^{\u03f5}\mathrm{(}v\mathrm{)}}},\text{\hspace{1em}(6)}$$(6)

$$L\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\displaystyle {\sum}_{l\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{l}_{\text{min}}}^{{N}_{l}}l{P}^{\u03f5}\mathrm{(}l\mathrm{)}}}{{\displaystyle {\sum}_{l\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{l}_{\text{min}}}^{{N}_{l}}{P}^{\u03f5}\mathrm{(}l\mathrm{)}}},\text{\hspace{1em}(7)}$$(7)

$$TT\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\displaystyle {\sum}_{v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{v}_{\text{min}}}^{{N}_{v}}v{P}^{\u03f5}\mathrm{(}v\mathrm{)}}}{{\displaystyle {\sum}_{v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{v}_{\text{min}}}^{{N}_{v}}{P}^{\u03f5}\mathrm{(}v\mathrm{)}}},\text{\hspace{1em}(8)}$$(8)

On the other hand, the transitivity, *TRAN*, is a network topological parameter based on the distance matrix elements **R**^{ϵ}:

$$TRAN\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}\frac{{\displaystyle {\sum}_{j\mathrm{,}k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}{R}_{ij}^{\u03f5}{R}_{jk}^{\u03f5}{R}_{ki}^{\u03f5}}}{{\displaystyle {\sum}_{i\mathrm{,}j\mathrm{,}k\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}{R}_{ij}^{\u03f5}{R}_{kj}^{\u03f5}}}.\text{\hspace{1em}(9)}$$(9)

These parameters were studied systematically with change in water volume rate flow *q*. The results are shown in Figure 4. It is noted that these figures show a non-monotonic behaviour of the studied flow phenomenon, namely they confirm the effect of resonating self-organisation (Figs. 2l and 3d for *q*=0.137 l/min). This is related to the values of *q* for which the elongated large bubbles are stable. They occur together with a low number of residual small bubbles. The recurrence rate, *RR*, (Fig. 1a) shows that the system returns to the same state most frequently at *q* ≈ 0.06 and 0.014, whereas between these value, the system behaves differently. It is concluded that for *q*=0.06 and 0.014, the fluctuations of laser transmitions appear between a small number of states, while for *q* ≈ 0.10, the number of states available significantly increases. This conclusion is supported by determinism, *DET*, and laminarity, *LAM*. At *q* ≈ 0.06, we observe the minimum, which indicates that the system becomes more random and unpredictable. The maxima 1/*L*_{max} correspond to the dynamical divergence that increases for three values of *q*. The maxima 1/*L*_{max} coincide with the minima in *RR* conforming the instabilities by increasing available states. The occurrence of intermittency is more probable for reaching such maxima in the dynamical divergence (1/*L*_{max}). Particular attention should be payed on the trapping time parameter, *TT*. This measure signals the probability that the system will last in a single state. Once the clear increases in *TT* are observed, the corresponding cases analysed by the RP (Fig. 3a and d) are represented by larger and clear checkerboard. Interestingly, this is the measure that directly indicates about the shape stability of air slugs or elongated bubbles. This stability strongly depends on the change of water flow rate *q* (Fig. 4e). Then, the bubbles merge into one large elongated slug upon increase in the water flow rate. Finally, transitivity is a topological measure of RP network itself. This can be linked to spatial and temporal properties of the system. Surprisingly, the *TRAN* measure changes in a similar way to the trapping time, *TT*. Nonetheless, it should be noticed that a clear checkerboard formation leads to isolated square islands signalled by minimum in transitivity. Simultaneously, it is also possible to measure the cluster size of a square island. The smaller the size of square, the lower the value of transitivity. Consequently, for unstable slug and elongated bubble flows, the squares in the RP are irregular and of smaller sizes (Fig. 3c,e, and f), which is signalled by minimal *TRAN* (Fig. 4f). The multiple peaks in patterns (Fig. 4) are due to aggregation of small bubbles into larger bubbles and slugs.

Figure 4: RQA measures versus water volume flow rate *q*: (a) recurrence rate – *RR*; (b) determinism – *DET*; (c) laminarity – *LAM*; (d) divergence – *DIV*=1/*L*_{max}; (e) trapping time – *TT*; (f) transitivity (causality) – *TRAN*, respectively.

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