TY - JOUR

T1 - Structural stability on manifolds with boundary

AU - Robinson, Clark

PY - 1980

Y1 - 1980

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85015488585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015488585&partnerID=8YFLogxK

U2 - 10.1016/0022-0396(80)90083-2

DO - 10.1016/0022-0396(80)90083-2

M3 - Article

AN - SCOPUS:85015488585

SN - 0022-0396

VL - 37

SP - 1

EP - 11

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -