## 2 Poincare Plot and Their Features
### Symbolic Dynamics  
When dealing with a non-linear dynamical system such as the [PERSON] model in Equations (2)-(4) with a phase space represented in Figure 2, the \(n\)-dimensional phase space can be partitioned into a finite set of contiguous cells, and each of the cells is assigned to a symbol [25]. A trajectory in the phase space generates a sequence of symbols describing the dynamics of the system.  
In this research, rather than partitioning the \(n\)-dimensional space, we propose the partition of the 2D surface \(\Sigma\) representing the interception of the current phase space with a plane. A 2D representation of the Lorenz model is shown in Figure 4. The dynamics of this model is represented in terms of their components \(x(t)\), \(y(t)\) and \(z(t)\) and their representation in \(\Sigma\) can be approximated by using the Poincare plot as shown in Figure 4. Such a plot is constructed by using only the component \(x(t)\); however, it can recover important features of the underlying dynamical system. In practical applications, such as fault detection in rotating machinery using vibration signals, the current underlying dynamical system is generally unknown and in the best of cases only approximate models are provided. The only known information is the time series representing the vibration signal, in consequence, we are constrained to attain a useful system representation using this information. When dealing with a dynamical system as previously described, the surface \(\Sigma\) of the phase space can be partitioned into \(r\) disjoint subsets denoted \(p=\{P_{1},P_{2},P_{3},...,P_{r}\}\). Each subset is represented by a symbol \(r\in\{1,2,3,...,r\}=S\). An orbit in the phase space is represented by a sequence of visited symbols \(\mathbf{s}=\{s_{1},s_{2},s_{3},...,\}\)[26]. The surface can be arbitrarily subdivided into several cells, and the points of the Poincare plot located in each cell can be represented by the same symbol. Alternatives of feasible partitions are Voronoi tessellations [27], or fractal based tessellations [28]. The sequence of generated symbols can be used for constructing state machines using sliding block codes [29] where a window of length \(N_{w}\) is shifted to the right one symbol at a time for constructing a \(N_{w}\)-Markov machine useful for anomaly detection [25]. In a previous research concerning faults classification in gearboxes, the Poincare plot has been subdivided in a set of angular regions and each region has been represented by a symbol. An illustration of this concept is presented in Figure 5. A fragment of a trajectory in the Poincare plot is shown and each point in the trajectory is assigned to a symbol according to the location of each point. Each angular region corresponds to one symbol and the partition of the phase space includes 12 symbols: \(S=\{1,2,3,4,5,6,7,8,9,10,11,12\}\). In this example, the trajectory is represented by the set of symbols \(\mathbf{s}=\{2,2,3,2,2,1,11,8,8,8\}\). The count of points corresponding to each symbol provides a set of features that can enable fault classification. This is illustrated in Figure 6 where the Poincare plot  
Figure 3: Poincaré map corresponding to the intersection of the phase space with the surface \(\Sigma\).  
constructed with component \(x(t)\) for the Lorenz model is shown along the angular partition of \(\Sigma\). The histogram of symbols calculated for this partition is shown in Figure 6b.  
Figure 4: Bi-dimensional representation of Lorenz model. (**a**) Time evolution of \(x(t)\). (**b**) Time evolution of \(y(t)\). (**c**) Time evolution of \(z(t)\). (**d**) Poincaré plot for \(x(t)\). The lag used for the Poincaré plot is \(\tau=20\).  
Figure 5: Poincaré plot showing a fragment of component \(x(t)\) of [PERSON] model including the set of symbols assigned.