R. Bowen. !-Limit Sets for Axiom A Diffeomorphisms. Journal of Differential Equations, 18:333, 1975. Home/Search Document Not in Database Summary Related Articles Check |
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A Brief Survey of Issues Relating to the Reliability of Simulation .. - Hayes (Correct)
....is successful if before the iteration the trajectory has noise ffi 0 f , after the iteration it has noise ffi 1 f , and ffi 1 f ffi 0 f for some reasonable 2 (0; 1) 5) Otherwise the refinement iteration is unsuccessful. Shadowing was first discussed by Anosov [3] and Bowen [10], in relation to hyperbolic systems. In a 2 dimensional hyperbolic system, there are two special directions called the unstable (or expanding) and the stable (or contracting) directions, which may vary with time and generally are not orthogonal. Small perturbations along the stable direction decay ....
R. Bowen. !-Limit Sets for Axiom A Diffeomorphisms. Journal of Differential Equations, 18:333, 1975.
An Open Set Of Maps For Which Every Point Is Absolutely.. - Yuan, Yorke (2000) (1 citation) (Correct)
....is crucial that numerical trajectories stay close to, in other words, they are shadowed (see Section 2 for definition) by, true trajectories; otherwise, the meaning of numerical results is far from obvious. Although compact hyperbolic invariant sets are shadowable as proved by Anosov [1] and Bowen [2], virtually all chaotic attractors that scientists encounter are nonhyperbolic. For example, the H enon strange attractors constructed by Benedicks and 1991 Mathematics Subject Classification. Primary h58F13i; Secondary h58F12, 58F14, 58F15i. This research was supported by the National Science ....
Rufus Bowen. !-limit sets for Axiom A diffeomorphisms. Journal of Differential Equations, 18(2):333--339, 1975.
Hyperbolic Sets for Noninvertible Maps and Relations - Sander (1996) (Correct)
....mapping to the same point, and thus many unstable manifolds. However, restricting to forward and backwards orbits, a similar theorem still holds for noninvertible maps and relations, as described in Chapter 4. Another result for hyperbolic sets for diffeomorphisms is the shadowing lemma of Bowen [5]. It pertains to pseudo orbits; these are sequences of points such that the image of each point in the sequence is no more than a small previously specified distance from the next point in the sequence. The shadowing lemma says that near a compact invariant hyperbolic set, every bi infinite ....
....with no unstable directions. Thus the entire space has hyperbolic structure. 4.5 Shadowing The shadowing lemma states that near a hyperbolic set, making small errors on each iteration still gives a reasonable picture of the dynamics. The shadowing lemma for diffeomorphisms is due to Bowen [5]. Here we give a proof of it for hyperbolic sets for smooth relations. The proof is functional analytic in nature. However, it seems to be conceptually simpler than the standard functional analytic proofs for diffeomorphisms, as the definition of hyperbolic sets in terms of a contraction allows ....
Rufus Bowen, !-limit sets for Axiom A diffeomorphisms, J. Diff. Equat. 18 (1975) 333-339.
A Fast Shadowing Algorithm for High Dimensional ODE Systems - Hayes, Jackson (1996) (Correct)
....iteration the trajectory has noise ffi 0 f , after the iteration it has noise ffi 1 f , and ffi 1 f ffi 0 f for some reasonable 2 (0; 1) 1) Otherwise the refinement iteration is unsuccessful. Shadowing was first discussed in relation to hyperbolic systems by Anosov [1] and Bowen [3]. In a 2 dimensional hyperbolic system, there are two special directions called the unstable (or expanding) and the stable (or contracting) directions, which may vary with time and generally are not orthogonal. Small perturbations along the stable direction decay exponentially in forward time, ....
R. Bowen. !-Limit Sets for Axiom A Diffeomorphisms. Journal of Differential Equations, 18:333, 1975.
Heteroclinic Networks in Coupled Cell Systems - Ashwin, Field (1997) (Correct)
....sets of points in the basin of attraction of Sigma. In particular, under what circumstances can we find points x 2 W s ( Sigma) such that (x) R( Sigma) We remark that the problem of characterizing those ( Sigma; OE) which are representable as limit sets has been considered by Bowen [6] in the context of discrete continuous dynamical systems defined on (possibly disconnected) spaces. Example 3.1. The Guckenheimer Holmes network Sigma arises as the heteroclinic network of a Delta 3 oZ 3 equivariant cubic vector field F on R 3 . We recall that F depends on three real ....
R Bowen. !-limit Sets for Axiom A Diffeomorphisms. J. Diff. Eqns., 18 (1975), 333--339.
Shadowing the Gravitational N-body Problem - Hayes (Correct)
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R. Bowen. !-Limit Sets for Axiom A Diffeomorphisms. Journal of Differential Equations, 18:333, 1975.
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