Analysis of intersections of trajectories of systems of linear fractional differential equations

Abstract

Present article deals with trajectorial intersections in linear fractional systems ('systems'). We propose a classification of intersections of trajectories in three classes viz. trajectories intersecting at same time(EIST), trajectories intersecting at distinct times(EIDT) and self intersections of a trajectory. We prove a generalization of separation theorem for the case of linear fractional systems. This result proves existence of EIST. Based on the presence of EIST, systems are further classified in two types; Type I and Type II systems, which are analyzed further for EIDT. Besides constant solutions and limit-cycle behavior, a fractional trajectory can have nodal or cuspoidal intersections with itself. We give a necessary and sufficient condition for a trajectory to have such types of intersections.and studies dynamics of its solution (referred as trajectory) for 0 < t < ∞.

Nonexistence of invariant manifolds in fractional order dynamical systems

“…e.g. The solution trajectories of classical differential dynamical systems are smooth whereas those of FOS can have self-intersections [52,53]. Some other differences are given in [54,55,56].…”