Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems

Abstract

The main purpose of this paper is to present a solution to the well-known problems generated by classical control methods through the analysis of nonlinear time series. Among the problems analyzed, for which an explanation has been sought for a long time, we list the significant reduction in control power and the identification of unstable periodic orbits (UPOs) in chaotic time series. To accurately identify the type of behavior of complex systems, a new solution is presented that involves a method of two-dimensional representation specific to the graphical point of view, and in particular the recurrence plot (RP). An example of the issue studied is presented by applying the recurrence graph to identify the UPO in a chaotic attractor. To identify a certain type of behavior in the numerical data of chaotic systems, nonlinear time series will be used, as a novelty element, to locate unstable periodic orbits. Another area of use for the theories presented above, following the application of these methods, is related to the control of chaotic dynamical systems by using RP in control techniques. Thus, the authors’ contributions are outlined by using the recurrence graph, which is used to identify the UPO from a chaotic attractor, in the control techniques that modify a system variable. These control techniques are part of the closed loop or feedback strategies that describe control as a function of the current state of the UPO stabilization system. To exemplify the advantages of the methods presented above, the use of the recurrence graph in the control of a buck converter through the application of a phase difference signal was analyzed. The study on the command of a direct current motor using a buck converter shows, through a final concrete application, the advantages of using these analysis methods in controlling dynamic systems.