In application of dynamical systems, we are often interested in the motion in a compact subset of Euclidean space or possibly of a torus cross a Euclidean space. The natural way to study these problems is in a manifold with boundary, e.g., a disk. This paper gives sufficient conditions for a C1 flow ƒt on a compact manifold with boundary, M, to be weakly structurally stable. In particular, if ƒ has a hyperbolic chain-recurrent set (Axiom A) and satisfies the transversality condition then it is weakly structurally stable. Since we impose no assumptions on the type of tangencies of trajectories with the boundary of M, we need to weaken the notion of conjugacy to allow a homeomorphism h from M into a collared manifold M′ which contains M in its interior. Such a homeomorphism allows us to bypass the singularity theory used by Percell and Sotomayor. A similar result for diffeomorphisms of manifolds with boundary is obtained. If the flow satisfies a quadratic external boundary condition, then it is possible to demand that the conjugacy preserve the boundary. These results also allow an infinite chain-recurrent set rather than the finite set of periodic orbits allowed in the previous results on manifolds with boundary.
ASJC Scopus subject areas
- Applied Mathematics