Kinematic N-expansive continuous dynamical systems

Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic [Formula: see text]-expansiveness for flows on a [Formula: see text] compact connected manifold [Formula: see text], which is an extension of [Formula: see text]-expansive homeomorphisms. We prove that if a vector field [Formula: see text] on [Formula: see text] is [Formula: see text] robustly kinematic [Formula: see text]-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic [Formula: see text]-expansive property; then, we study their robustness.