# A Nonlinear Method for Characterizing Discrete Defects in Thick Multilayer Composites
## 2 Methodology  
If the data is aperiodic and does not recognize simple rules of their time dependence, then an approximate repetition of certain events, called a recurrence, can help us build more complex rules [18]. The recurrence analysis of the dynamics of a system is conducted in a phase space that was constructed with delayed vectors. A sequence of scalar measurements \(x(t_{n}),\ n=1,2,\ldots,N\) can be extended to a vector by the Takens delay method [19]:  
\[\overrightarrow{x}_{n}=(x_{n-(m-1)\tau},x_{n-(m-2)\tau},\cdots,x_{n-\tau},x_{n}) \tag{1}\]  
where \(\tau\) is the time lag and \(m\) is the embedding dimension. The values of \(m\) and \(\tau\) determine the fact of whether the required information can be obtained from the original time series, while improper parameters will seriously distort the analysis seriously. The common selection method of embedded dimension \(m\) is based on false nearest neighborhood, and the selection method of delay time \(\tau\) is the average mutual information method, according to reference [20]. A method for visualizing recurrences is called a recurrence plot (RP) and [PERSON] et al. have introduced it [21]. Compute the matrix  
\[R_{i,j}=\Theta\big{(}\varepsilon-\|x_{i}-x_{j}\|\big{)},i,j=1,2,\ldots,n-(m-1)\cdot\tau \tag{2}\]  
where \(\Theta\) is the Heaviside step function (i.e., \(\Theta(x)=0\) if \(x<0\), and \(\Theta(x)=1\) otherwise), \(\|\bullet\|\) is the Euclidean distance between the two vectors, \(\varepsilon\) is a tolerance parameter to be chosen, and \(x_{i}\) is the delayed vectors of some embedding dimension. Darken all of the nonzero values in the recurrence matrix \(R_{i,j}\) and the RP is attained, as shown in Figure 2, in which many special structures exist. According to the macroscopic structures of the RP, the characteristics of plots refer to the different dynamics of system. Figure 2a is a homogeneous RP, which is shown as a uniform distribution of single recurrence dots, and it represents a typical stationary system, such as a random time series. Figure 2b is a period RP, which is shown as a long diagonal structure, and it represents a periodic oscillation system. Figure 2c is the RP of a chaotic system, shorter diagonal lines, small blocks, and single dots can be found as the suggest of chaos [22]. There are also microscopic features, such as single dots, diagonal lines, and vertical and horizontal lines in the RP. The appearance of single dots indicates that the corresponding state does not last or greatly fluctuates. Diagonal lines consist of a series of adjacent recurrence dots, and most of them are parallel to the main diagonal line. A diagonal line represent that the system track is similar in the same direction within a certain time period, and its length represents the degree of determination or predictability. Vertical or horizontal lines represent time segments that remain unchanged or change very slowly, and they are typical behaviors of the state of the laminate, which can reveal the discontinuity of the signal. Usually, the macrostructure of RP can help us to directly observe the differences in the general structure of the system, while the results are significantly affected by the individual subjective judgment. Therefore, it is necessary to conduct a quantitative analysis that is based on microstructure. The statistical method RQA is more persuasive. In general, RP analysis provides a visual inspection of the matrix in Equation (2), and RQA analysis provides statistical variables that are based on diagonal, vertical, or horizontal lines formed by recurrence dots in the matrix. Based on the diagonal lines in the RP, [PERSON] and [PERSON] put forward some quantities to measure the complexity of the system [23]:  
1. Recurrence rate (RR) \[RR(\epsilon)=\frac{1}{N^{2}-N}\sum\limits_{i\
eq j=1}^{N}R_{i,j}(\epsilon)\] (3) counts the black dots in the RP excluding the main diagonal line. \(RR\) is a measure of the relative density of recurrence points in the recurrence matrix. 2.