On the stability of bimodal systems in ℝ3

In this work we investigate the stability of bimodal and bistable (both modes are stable) continuous time linear systems in ¿³. Under certain conditions, we first show that all the trajectories which start on the plane separating the two modes are bound to hit the plane again in finite time and go into the other mode. This property yields fixed directions on the separating plane. Eventually, all the trajectories start hitting the separating plane arbitrarily close to the fixed directions. Finally, it is proven that the over all system is stable if and only if a trajectory starting from a fixed direction is stable.