Global convergence of nonlinear cascade flows with Morse–Bott zero dynamics

We derive a sufficient condition that a flow captures the dynamics on an invariant submanifold. This leads to a refinement of the LaSalle invariance principle. As a consequence, we generalize a well-known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set. This includes the case where a globally asymptotically stable system is coupled to a Morse–Bott flow.