# Analysis of Dynamic Response of a Two Degrees of Freedom (2-DOF) Ball Bearing Nonlinear Model
## 4 Recurrence Analysis
### Recurrence Quantification Analysis (RQA)
#### 4.2.4. Quantitative Based on the Probability  
* Recurrence period density entropy (\(T_{rec}\))--quantifies the extent of recurrences [84,85]: \[T_{rec}=-\sum_{i=1}^{T_{max}}P(i)lnP(i),\] (26)
* Clustering coefficient (C)--represents the probability that two recurrences of any state are also neighbors [36,86]: \[C=\frac{\sum_{i,j=1}^{N}A_{v,j}A_{ij}A_{ij,v}}{k_{v}(k_{v}-1)},\] (27)
* Transitivity (\(TRANS\))--is the measure allowing to differentiate the periodic or chaotic dynamics [85]: \[T=\frac{\sum_{i,j,k=1}^{N}R_{ij,k}R_{i,j}R_{i,k}}{\sum_{i,j,k=1}^{N}R_{i,j}R_ {i,k}\Big{(}1-\delta_{j,k}\Big{)}},\] (28)  
The bearing's dynamic response was analyzed with the above-described quantificators, to estimate which of them are sensitive to the variable eccentricity parameter. The results are visualized for the eccentricity \(ecc=\{0;0.4\}\) with the variable step of 0.05 in Figure 8. The fixed recurrence rate was employed for the calculation of each quantificator to obtain consistent results. Figure 8: \(Cont.\)  
The determinism refers to the system's predictability and its results are convergent with the character of obtained recurrence plots. _DET_ is relatively high (close to 1) for all values of the eccentricity and it slightly decreases with larger eccentricity, then the nonlinear effects influence at most the system's dynamics. Figure 8: Results of Recurrence Quantification Analysis (RQA) for the series of rotor’s eccentricity. The analyzed quantificators are following: (**a**) determinism, (**b**) averaged diagonal length, (**c**) length of the longest diagonal, (**d**) entropy of the diagonal length, (**e**) laminarity, (**f**) trapping time, (**g**) length of the longest vertical line, (**h**) recurrence time of the 1 st type, (**i**) recurrence time of the 2 nd type, (**j**) recurrence period density entropy, (**k**) clustering coefficient, (**l**) transitivity. The average length of the diagonal line gives similar information on the system's dynamics as the determinism and their trends are very up-close to each other. However, \(L\) indicates the stability of periodic intervals reflected in the recurrence plots in a more veritable way, especially it decreases for the high values of the eccentricity when the chaotic dynamics is present from the \(ecc=0.25\). Interestingly, the quanticator length of the longest diagonal \(L_{max}\) has reversed character to the length of the average diagonal \(L\). The divergence of the \(L_{max}\) values corresponds to the broadening effect of the central diagonal line (see Figure 7). In such a case, \(L\) should be more considered as a more reliable indicator of periodic behavior deficiencies. Entropy provides information on the uncertainty of the bearing's response. Clearly, the value of entropy is increasing with the increase of the eccentricity. For the periodic solutions, its value is at a constant level. In the case of the laminarity, it refers to the meantime, when the state of the system is trapped in some states. It also indicates switching between different states of the system. Note, that the laminarity value is close to one in a wide range of eccentricity. As the analyzed mathematical model is deterministic, the transient states can appear only in chaotic response. Consequently, the vertical lines are fairly short, which is also reflected by a small value of the \(TT\) parameter. Trapping time refers to the average length of the vertical lines by the measurement time scale of small changes in the response. The change of \(TT\) is convergent with changes observed at recurrence plots, its increasing trend corresponds to the level of the chaotic response of the bearing. The \(V_{max}\) has a very similar character to the \(TT\), square-like structures form longer and longer vertical lines with increasing eccentricity. This quanticator strongly reflects obtained structures of RPs. Recurrence time is mostly used for stochastic signals with numerous transient states, so those measures are not practical for considering the mathematical model.