STRUCTURE AND STABILITY OF BIMODAL SYSTEMS IN R^3: PART 1

In this paper, the structure and global asymptotic stability of bimodal systems in R^3 are investigated under a set of assumptions which simplify the geometric structure. It is basically shown that one of the assumptions being used reduces the stability problem in R3 to the stability problem in R^2. However, structural analysis shows that the behavior of the trajectories changes radically upon the change of the parameters of individual subsystems. The approach taken is based on the classification of the trajectories of bimodal systems as i) the trajectories which change modes finite number of times as t → ∞, and ii) the trajectories which change modes infinite number of times as t → ∞. Finally, it is noted that this approach can be used without some of the assumptions for all bimodal systems in R^3, and for bimodal systems in R^n