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

This article deals with trajectorial intersections in systems of linear fractional differential equations. We propose a classification of intersections of trajectories into three classes: (a) trajectories intersecting at the same time (IST), (b) trajectories intersecting at different times (IDT), and (c) self-intersections of a trajectory. We prove a generalization of the separation theorem for the case of linear fractional systems. This result proves the existence of the IST. Based on the presence of the IST, systems are further classified into two types, Type I and Type II systems, which are analyzed further for the IDT. Self-intersections in a fractional trajectory can be regular such as constant solution or limit-cycle behavior, or they can be irregular such as cusps or nodes. We give necessary and sufficient conditions for a trajectory to be regular.