Morse-Smale systems From Scholarpedia Michael Shub (2007), Scholarpedia, 2(3):1785. doi:10.4249/scholarpedia.1785 revision #73466 [link to/cite this article] Hosting and maintenance of this article is sponsored by Brain Corporation. Curator: Dr. Michael Shub, University of Toronto, CANADA More-Smale systems are the simplest dynamical systems. They are structurally stable and have intimate connections to the topology of manifolds. We elucidate these concepts below. Contents 1 Dynamical Systems 2 Periodic orbits 3 Hyperbolicity 4 Stable and unstable manifolds 5 Morse-Smale Dynamical systems: Definition 6 Topological Conjugacy 7 Structural Stability 8 Morse-Smale Gradient Fields and Relations to Topology 8.1 Example 9 Morse-Smale Diffeomorphisms: Examples and Relations to Topology 10 Bibliography 11 External links 12 See Also Dynamical Systems By a dynamical system we mean here a , diffeomorphism of a compact differentiable manifold without boundary or a one parameter group such that , and the vector field defined by the tangents to the orbits is well defined and .