## 3 Results and Discussion
This method allows for distinguishing between regular and chaotic behavior based on time series data without requiring phase space reconstruction. It is particularly useful in systems where the nature of oscillations changes with varying parameters, such as airflow velocity or structural configuration [55; 56]. The test provides a quantitative measure (the \(K\)-value) indicating the presence of deterministic chaos (\(K\approx 1\)) or regular motion (\(K\approx 0\)). The coefficient \(K\) can be calculated using two main approaches: the mean square displacement (MSD) method and the covariance-based method. In the MSD approach, the displacement of a transformed time series is analyzed as a function of time; for chaotic \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Diameter** & \\
**Mass** & **20 mm** & **25 mm** & **30 mm** \\  & **Threshold** & **Threshold** & **Threshold** \\  & \(U\)**[m/s]** & \(U\)**[m/s]** & \(U\)**[m/s]** \\ \hline \(m_{1}\) [g] & 9 & 8 & 6 \\ \(m_{2}\) [g] & 9 & 7 & 6 \\ \(m_{3}\) [g] & 10 & 9 & 6 \\ \(m_{4}\) [g] & N/D & 8 & 6 \\ \(m_{5}\) [g] & N/D & 6 & 6 \\ \hline \hline \end{tabular} N/D—not detected.  
\end{table} 
Table 3: Identified transition points between oscillatory and transient regimes. systems, the MSD grows linearly, while for regular systems, it saturates at a constant level. The formula for calculating MSD is as follows [57; 58]:  
\[M(n)=\frac{1}{N-n}\sum_{j=1}^{N-n}\Bigl{[}(p(j+n)-p(j))^{2}+(q(j+n)-q(j))^{2} \Bigr{]} \tag{5}\]  
where  
\(N\)--total number of data points in the time series,  
\(n\)--time lag or displacement index, varying from 1 to a chosen maximum,  
\(j\)--summation index ranging from 1 to \(N\)-\(n\),  
\(p(j)\), \(q(j)\)--transformed coordinates in the 2D phase space. The quantity \(M(n)\) captures how far the point (\(p(n)\),\(q(n)\)) \"diffuses\" over time. For regular systems, \(M(n)\) remains bounded or grows slowly. For chaotic systems, \(M(n)\) grows linearly with \(n\). The covariance-based approach in the 0-1 test calculates the \(K\) value by analyzing the asymptotic growth rate of the variance of a transformed time series. In chaotic systems, the transformed trajectory behaves similarly to Brownian motion, resulting in a linearly increasing variance and a \(K\) value close to 1. In contrast, for regular systems, the trajectory remains bounded, leading to sublinear growth of variance and a \(K\) value close to 0. To evaluate the growth trend of the mean square displacement \(M(n)\), the Pearson correlation coefficient \(K(c)\) is computed between \(M(n)\) and the time lag \(n\), and it is given by [59]:  
\[K(c)=\frac{cov(n,M_{c}(n))}{\sqrt{cov(n,n)*cov(M_{c}(n)M_{c}(n))}} \tag{6}\]  
where  
_cov_--denotes covariance operator,  
\(M_{c}(n)\)--centered mean square displacement,  
\(n\)--time lag. In the proposed approach to applying the 0-1 test, both methods were used, comparing the obtained value of the \(K\) indicator (Figure 8). The expected results are inversely proportional to those obtained for the average RMS (Figure 7). In cases of the highest voltage values, the \(K\)-value remains low, only slightly above 0 up to 0.3, indicating regular oscillations of the system with minor noise impact. This situation changes when irregular, chaotic vibrations occur; then, an increase in the \(K\)-value is expected, approaching 1, which indicates chaotic behavior. In the conducted analysis, the \(K\)-value was divided into two types: \(K1\), calculated based on the mean square displacement (MSD) (Figure 8a,c,e), and \(K2\), determined using the Pearson correlation coefficient (Figure 8b,d,f). This distinction allows us to capture different aspects of the system's dynamics: \(K1\) reflects the asymptotic growth behavior, while \(K2\) quantifies the linear correlation structure in the transformed phase space. In the case of the MSD approach, the effectiveness of the method is less noticeable for the smallest diameter, 20 mm, due to the presence of oscillations across nearly the entire range of airflow velocities.