## 2 Poincare Plot and Their Features  
### Poincare Plot  
The Poincare plot is related to the Poincare map that is used for studying non-linear dynamics of chaotic systems [21]. A physical system can be represented as a set of first-order differential equations:  
\[\dot{x}=\frac{dx}{dt}=f(x),\qquad x\in\mathbb{R}^{n}, \tag{1}\]  
where \(f\) is a smooth vector field. The solution for this set of equations is denoted \(\phi(x,t)\), that can satisfy \(\phi(x,0)=x\). The expression \(o(x)=\{\phi(x,t)/x\in\mathbb{R}^{n}\}\) is known as an orbit or trajectory in the phase space. The Poincare map is obtained by considering successive intersections of a trajectory with a lower-dimensional surface denoted \(\Sigma\) of the phase space \(\mathbb{R}^{n}\) as shown in Figure 1.  
In general, the Poincare map is well defined when the system has periodic solutions. In this case there exists a positive number \(T\) representing the orbit period. Similarly, the system has an initial solution denoted \(x^{(0)}\). The solution satisfies \(\phi(x^{(0)},T+t)=\phi(x^{(0)},t)\) for all \(t\in\mathbb{R}\). Particularly,  
Figure 1: Trajectories and Poincaré map created in \(\Sigma\). \(\phi(x^{(0)},T)=\phi(x^{(0)},t)=x^{(0)}\) then the point \(x^{(0)}\) returns to himself after flowing during the time \(T\). The Poincare map reduces the dimension of the problem by one because the system can be studied by considering only the points included in \(\Sigma\). Unfortunately, except for trivial cases, the Poincare map cannot be expressed explicitly [16]. In general, the Poincare map construction requires solving the set of differential equations and detecting the points where the orbits intercept the surface \(\Sigma\). The Poincare map has the property of reflecting characteristic features of the flow \(\phi(x,t)\). For instance, stable or regular trajectories result in simple curves when they are represented in the map. In contrast, unstable or chaotic trajectories result in sequences of random points in the Poincare map. In practical engineering applications, we deal with complex problems where accurate knowledge about the physical relations of different parts of the system is not available. There is not any mathematical model for such a system, and sometimes we only have a measurement of a given variable as a function of time, such as a vibration signal denoted \(x(t)\). In these cases, the Poincare map cannot be calculated. Then, approximate solutions are provided such as plotting the estimated first derivative of the time series \(dx(t)/dt\) with respect to \(x(t)\)[22], or plotting each value of the time series \(x(t)\) with respect to their delayed version \(x(t-d)\), or equivalently \(x(t+d)\) versus \(x(t)\)[21]. This type of plot is known as delay-coordinate reconstruction, delay plot or, in the domain of application of biomedical engineering, it is known as the Poincare plot [23]. An illustration of this concept is shown in Figure 2, where the 3D representation of the Lorenz attractor is depicted in terms of the phase-space trajectories [24]. The Lorenz model was developed for studying atmospheric convection. However, their application has been extended to other domains such as electric circuits and brushless DC motors. The Lorenz model is composed by a set of three ordinary differential equations:  
\[\frac{dx}{dt}=\sigma(y-x) \tag{2}\]  
\[\frac{dy}{dt}=x(\rho-z)-y \tag{3}\]  
\[\frac{dz}{dt}=xy-\beta z \tag{4}\]  
where the parameters are the Rayleigh number \(\rho\) or, the Prandtl number \(\sigma\), and an additional parameter \(\beta\). Typical values are \(\rho=28\), \(\sigma=10\) and \(\beta=8/3\). Given an initial point \((x_{0},y_{0},z_{0})\), the system of equations is solved and the evolution of each of the variables is represented as a 3D phase space. Such phase space is intersected with a surface \(\Sigma\) representing a plane.