On almost orbital equivalence of nonlinear systems

This paper is concerned with equivalence of nonlinear systems from a viewpoint of geometric congruence of system orbits in the state space. The notion of orbital equivalence had been originally exploited by Sampei and Furuta in the context of the time-scale transformation approach. They gave a fundamental characterization of orbital equivalence in the form of a similarity (parallelism) of the system vector fields; however, it was not satisfactory in the sense that there remains significant gap between the necessary condition and the sufficient one, mainly due to its treatment of unavoidable singularity of the time-scale functions. In this paper, we intend to cast a new light on this problem with the aid of almost point of view that neglects the inconsistency on measure zero subsets. By introducing the notions of almost orbital equivalence and the almost similarity of the vector fields, we provide the sole necessary and sufficient conditions connecting them. Furthermore, we discuss the combination of the orbital equivalence analysis and stability issue based on the recent development of the density function approach for stability analysis. The discussion is also illustrated by numerical examples.