# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 4 Recurrence Analysis
### Recurrence Plots (RPs) Method  
The forerunners of the graphical interpretation of the considered distance matrix \([R]\) are [PERSON] et al. [38] and [PERSON] et al. [39], who proposed its analysis in the form of recurrence plots. The mathematical relationship (Equation (12)) describes the formation of the distance matrix with recurrence points and empty spaces (lack of recurrence):  
\[\left[R_{i,j}\right]=\left\{ \begin{array}{l}1:\ \{x_{i}\}\approx\left\{x_{j} \right\}\\ 0:\ \{x_{i}\}\
eq\left\{x_{j}\right\}\end{array} \right.,\ i,j,\ldots,\ N, \tag{12}\]  
where \(\{x_{i}\}\approx\{x_{j}\}\) are the points in the close distance appointed by the threshold radius of \(\varepsilon\) creating a recurrence point. Before the recurrence plots can be created, the three parameters must be found beforehand, i.e., time delay--\(\tau\), embedding dimension--\(m\), and threshold--\(\varepsilon\). According to [PERSON] theorem [72], three mentioned parameters are demanded creating the missing coordinates, then the state of the system after reconstruction can be represented in form of a time-delayed vector [32]:  
\[\underset{x\ i}{\rightarrow}=\left[x_{i},x_{i-\tau},x_{i-2\tau},\ldots,x_{i- (m-1)\tau}\right], \tag{13}\]  
The first parameter to be found for the phase space reconstruction is the time delay \(\tau\) for which two methods are specified (a) autocorrelation function and (b) mutual information. The first method is the autocorrelation function [73] given by the following formula:  
\[c_{\tau}=\frac{s_{n}s_{n-\tau}}{\sigma^{2}}, \tag{14}\]  
where, \(n\) is the time index of the dynamical process, \(\tau\) is the time delay, \(\sigma^{2}\) is the variance of the considered time series. Here the characteristic \(\tau\) is found to be a decay of \(c_{\tau}\) reaching 0 or the first minimum. Another approach for the definition of the time lag is the mutual information function (MI) [74; 75]. The method is based on the quantification between the original time series and delayed time series (shifted), the value of time lag \(\tau\) for the phase space reconstruction is the first minimum. In Figure 5(a), the exemplary mutual information function is presented (\(ccc\) = 0.4) with the marked first minimum (red-dashed line). The mathematical description of \(MI\) has the following form:  
\[I(x(t),x(t+\tau))=\sum_{i,j}p_{i,j}(\tau)\log\left(\frac{p_{i,j}(\tau)}{p_{i} p_{j}}\right), \tag{15}\]  
where \(I(x(t),x(t+\tau))\) is the mutual information function between the original signal and delayed time series, \(p_{i}\), \(p_{j}\) are the probability that \(x(t)\) is in bin \(i,j\) of the histogram constructed from the data points in \(x\), \(p_{ij}(\tau)\) is the probability that \(x(t)\) is in bin \(i\) and \(x(t+\tau)\) is in bin \(j\). The most popular method of determining the embedding dimension \(m\) was proposed by [PERSON] et al. [76] based on the \"False Nearest Neighbors\" function. The FNN function detects points in a close distance to each other in the embedding space. The number of embedding dimension \(m\) is determined by zero of the FNN function (Figure 5(b)), then all false neighbors disappear and no further increase of the dimension is necessary [77]. The most demanding step in the phase-space reconstruction is the choice of the threshold \(\varepsilon\) corresponding to the radius in the phase space. [PERSON] in his work [35; 78]collected a rich source of knowledge on the criteria for its selection, however, the applied method is mostly dependent on the analyzed dynamical system. For our analysis, we assume for the calculation, the method of fixed recurrence rate of 5%. Then the threshold value is adjusted to the density of the recurrence points at RP for all considered cases. This method has two following advantages:  
1. [label=(0)]
2. Considered dynamical states are dependent on only one feature at a constant level. 3. There is no need to normalize considered time series before the phase-space reconstruction.