Basic structure of the dynamics with saturation limits on states

This paper studies some fundamental properties of the large nonlinear differential system with saturation limits on states. It is shown that for such systems: (1) no Lipschitz condition is satisfied; (2) multiple reduced order flows exist on the limit surfaces; (3) trajectories are only unique forward in time but not backward in time. New equilibrium points induced by state limits are identified. Furthermore, the Jacobian conditions on the stability and local invariant manifolds of these unconventional equilibria are established. The structural complexity for global invariant "manifolds" and the impact of state limits on the region of attraction are also examined in this paper.