A phase space is a space in which all states of a system are represented and a phase portrait is a visual representation of the trajectory of this space. For the two-dimensional case, the phase space will turn into a phase plane. The phase plane trajectory consists of a closed single loop for a periodic response and multiple loops for a multi-periodic behavior. The topology of the phase space trajectory provides valuable information regarding the behavior of a system in a qualitative fashion. The method of Phase Space Topology (PST) which was developed by our team at Villanova University is a technique for characterizing this topology in the periodic and multi-periodic domain with quantitative measures. PST quantifies the topology of these closed curves by computing the density of points along each axis of the phase portrait. For simplicity of illustration, the examples here are presented in the two dimensional space; however, it can be extended to higher dimensions. For dimensions higher than three, even though the visualization of the phase space trajectory is not possible, the method is still applicable. In fact, the computations are performed individually and independently for each state of the system.

To illustrate the method, consider the three sample phase portraits (position x1 vs. velocity x2 of the first mass) shown below along with the corresponding estimated density of x1 and x2 time series. The data has been obtained from a MDOF nonlinear mass-spring system. Let us now see how the density of x1 and its peaks properties change based on the topology of the phase portraits. In Fig. a, we have a double-loop phase portrait which is a characteristics of a bi-periodic motion. The edges of the loops at the returning points in x1 direction have produced four sharp peaks in the density plot of x1. The two sharp peaks in the middle are higher than the other two due the lower radius of the phase trajectory in those areas. In addition, the depression of the phase trajectory in the middle has produced a smoother peak in the density plot. In Fig. b, we have a similar topology; however, the depressed part has moved rightward. It can be seen from the corresponding density plot that the smooth peak has shifted rightward to the middle of the two sharp peaks as well. In Fig. c another loop has evolved in the phase plane trajectory; representing a response with three frequencies. As a result, two more sharp peaks have emerged in the density plot of x1. The density plots of x2 can also be explained in a similar way.

Sample phase portraits of the system response along with the density plots of the corresponding states

This density function can be computed and plotted for any state of the system. According to the figures, the shape of the phase space trajectory which is a closed curve for periodic and multi-periodic motions is in a direct relationship with the properties of the peaks in the density plots. Each peak in the density plots can be characterized with its location, height, and sharpness values. These values can be used as effective features in order to train a neural network and estimate the defective parameters of the system.