Attractors: Persistence, and Density of Their Basins
- 1 January 1982
- journal article
- Published by JSTOR in Transactions of the American Mathematical Society
- Vol. 269 (1) , 247-271
- https://doi.org/10.2307/1998602
Abstract
An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors. A quasi-attractor is any nonempty intersection of attractors. It is shown that quasi-attractors other than attractors occur for a large set of flows. It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows. Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor". Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.Keywords
This publication has 20 references indexed in Scilit:
- Structural Stability and Hyperbolic AttractorsTransactions of the American Mathematical Society, 1979
- A modification of the Kupka-Smale theorem and smooth invariant manifolds of dynamical systemsMathematische Nachrichten, 1979
- Isolated Invariant Sets and the Morse IndexCBMS Regional Conference Series in Mathematics, 1978
- The Hopf Bifurcation and Its ApplicationsJournal of Applied Mechanics, 1978
- The periodic points of maps of the disk and the intervalTopology, 1976
- Differential TopologyPublished by Springer Nature ,1976
- The ergodic theory of AxiomA flowsInventiones Mathematicae, 1975
- Equilibrium States and the Ergodic Theory of Anosov DiffeomorphismsPublished by Springer Nature ,1975
- Filtrations, Decompositions, and ExplosionsAmerican Journal of Mathematics, 1975
- Invariant manifoldsBulletin of the American Mathematical Society, 1970