**2. Recurrence plot**

The basis for constructing recurrence plots is the phase diagram or phase plane. The phase plane [38] describes the stability of a dynamic system. It determines the relationship between the potential and kinetic energies at a given time interval and predicts the evolution of the energy balance. The recurrence plot is constructed from the analysis of the evolution of a phase plane. The dynamic system evolves following different paths represented in phase planes; these planes are built at intervals defined by the fundamental period. Then the recurrence plot identifies the number of times that a phase plane vector (the state variables that describe the system dynamics) repeats or recurs at every fundamental period. Before describing the construction of a recurrence plot, the following section introduces the basis of the phase plane.

#### **2.1 Phase plane**

A phase plane describes the mass trajectory along a two-dimensional energy field. (Two-dimensional state variables). Representing the dynamic behavior of a linear system as a differential equation system:

$$
\dot{\mathbf{x}} = \mathbf{B} \mathbf{x} \tag{1}
$$

Matrix *B* contains the system's parameters, and it is derived from Hamilton's principle:

$$H(p,q) = T(p) + V(q) \tag{2}$$

Or

$$T(p) = \frac{p^2}{2m} \tag{3}$$

Hamilton's principle states that the equilibrium of the system is obtained when:

$$
\dot{q} = \frac{\partial H}{\partial p} \tag{4}
$$

And

$$
\dot{p} = -\frac{\partial H}{\partial q} \tag{5}
$$

Therefore, at any instant, there is a function *ϕ*ð Þ *p*, *q* such that

$$\frac{d\phi}{dt} = \frac{\partial\phi}{\partial q}\frac{\partial H}{\partial p} - \frac{\partial\phi}{\partial p}\frac{\partial H}{\partial q} \tag{6}$$

The dynamic stability and the evolution of the phase plane can be derived from Eq. (6). If, in the phase plane, the function *ϕ* is constant in any trajectory, then the system is stable (Liouville's theorem).

$$\frac{dH}{dt} = \frac{\partial H}{\partial q}\dot{q} + \frac{\partial H}{\partial p}\dot{p} = 0\tag{7}$$

Eq. (7) implies that the phase plane has a constant volume.

If the mass is constant, the linear momentum only depends on the mass velocity. In linear systems, the potential energy is proportional to the displacement; therefore, the phase plane can be represented with the state variables velocity and position. **Figure 1** shows the phase plane of a single degree of freedom system with a harmonic response (*p*\_ <sup>þ</sup> *kq* <sup>¼</sup> <sup>0</sup> *and <sup>q</sup>*\_ <sup>¼</sup> *<sup>p</sup> m*Þ. It is clear that the trajectory is constant and smooth; meanwhile, a nonlinear system will show irregular patterns. These irregularities are the basis for studying nonlinear systems with the recurrence plots.

#### **2.2 Definition of the recurrence plot**

The phase plane function can be defined as a vector that depends on a parameter (time) and defines the state of the dynamic system (**Figure 1**):

$$\overline{\mathfrak{X}}(t) = [\overline{\mathfrak{X}}\_1, \overline{\mathfrak{X}}\_2, \dots, \overline{\mathfrak{X}}\_n] \tag{8}$$

The dynamic evolution in time is represented in **Figure 2**. If the dynamic response is steady, then *x*1ðÞ¼ *t xn*ð Þ *t* þ *nτ* , where *τ* is the period of a harmonic system and *n* is an integer.

A system recurs if two subsequent states are equal and it repeats every period *τ*. According to Eckmann, Kamphorst, and Ruelle [1], a recurrence plot is a matrix representation of the similarity of two consecutive state conditions:

*Perspective Chapter: Predicting Vehicle-Track Interaction with Recurrence Plots DOI: http://dx.doi.org/10.5772/intechopen.105752*

**Figure 1.** *Phase plane of a linear harmonic system.*

**Figure 2.** *Evolution of the state variables of a linear harmonic system.*

$$R\_{\vec{\eta}} = \begin{cases} \mathbf{1} : \overline{\mathbf{x}}\_{i} = \overline{\mathbf{x}}\_{\vec{\jmath}} \\ \mathbf{0} : \text{otherwise} \end{cases} i, j = \mathbf{1}, \dots, N \tag{9}$$

The procedure applies to a continuous and discrete set of data. But, in general, data are discrete vectors; thus, *N* is the number of state vectors in the time array, divided by loops of period *τ*. Data contain noise and truncation errors; therefore, two vectors cannot have the same magnitude, and Eq. (9) is modified as:

$$R\_{ij} = \begin{cases} \mathbf{1} : \left| \overline{\mathbf{x}}\_i - \overline{\mathbf{x}}\_j \right| < \varepsilon, \\ \mathbf{0} : \left| \overline{\mathbf{x}}\_i - \overline{\mathbf{x}}\_j \right| > \varepsilon, \end{cases} i, j = \mathbf{1}, \dots, N \tag{10}$$

where *ε* is a tolerance value. The tolerance should be less than 10% of the mean diameter of the phase plane or five times larger than the standard deviation of the observational noise.

The orbits in a phase plane describe the system's dynamics. Orbits can see growth due to the entropy, and the number of orbits depends on the characteristic frequencies, damping, and the presence of nonlinear behavior. These characteristics modify the recurrence plots in a way that they can describe the system's behavior and predict future states. There are several characteristics of a recurrence plot that are classified as topology and texture. These characteristics are related to the system's dynamics.

Since the recurrence plots depend on the phase plane, it is important to construct a proper phase plane. The phase shift principle helps to build the phase plane of singlefrequency signals; but for mechanical systems, this procedure introduces artificial noise that corrupts the phase plane [4]. The following sections describe the procedure for constructing the phase plane from acceleration measurements.

#### **2.3 Topology and texture**

In a Recurrence Plot, the main diagonal is always present, and parallel diagonals occur only in periodic or quasi-periodic systems. The distance between the diagonals is the fundamental period, and the diagonal lengths define if a system has a predictable behavior (steady condition). If the system presents recurrence but at different frequencies, then the diagonals will be shifted from the main diagonal.

Eckmann et al. [3] presented a summary of different topologies:


The topology depends on the tolerance, if *ε* is small, the Recurrence Plot erases most of the similar vectors, and it will be almost empty. Too large tolerance will produce many artificial figures that have no relation to the dynamic behavior.

#### **2.4 Recurrence plot characteristics**

Besides the topology and texture, a recurrence plot has several characteristics related to the system's dynamic response [39]. The following table (**Table 1**) summarizes the main features classified as recurrence quantification analysis.

The first step in constructing a recurrence plot is to build the phase plane. The problem with creating the phase plane using acceleration data is the integration of the

*Perspective Chapter: Predicting Vehicle-Track Interaction with Recurrence Plots DOI: http://dx.doi.org/10.5772/intechopen.105752*


**Table 1.** *Recurrence quantification parameters.*

state variables. The data are time series vectors *x*€, and the phase plane is built with the state variables *x*\_ and *x*. When the signal has only one frequency, the integration can be achieved by normalizing the original data and shifting them to one-quarter of the period. The following section describes the procedure for the integration of time series with any pattern.
