Pseudo Linear Systems: Stability Analysis and Limit Cycle Emergence

Abstract

Nonlinear autonomous systems, often arise in different fields of science, are usually difficult to analyze. Pseudo linear representation of such systems recently has become very popular to deal with this difficulty. This paper presents a deep through analysis about the stability of nonlinear autonomous systems represented by pseudo linear forms. By discretization of the continuous-time dynamical systems,
the stability of original nonlinear system is investigated via the discretized model and some new results are obtained for a class of pseudo linear systems. Due to the fact that the error between discrete model and that of the original continuous-time model vanishes as the sampling time goes to zero, in this paper we consider almost zero sampling time throughout of our analysis to make sure about the validity of
results obtained for the continuous-time nonlinear systems. Based on the discretized model, some conclusive propositions are established; this apparently provides a framework to tackle the long struggle in the stability consideration of nonlinear systems via pseudo linear form by applying the crucial role of
nonlinear eigenvectors neglected in the previous studies. In addition, based on these stability analysis results and the qualitative analysis tools provided through pseudo linear representation of nonlinear systems, the question of limit cycle emergence is also tackled. It is shown that, as an elegant application
of the proposed qualitative analysis tool, the generation of limit cycles with desired shape and numbers can be easily performed. Some illustrative examples are finally given to highlight the validity of the proposed analysis technique.