Robust chain transitive vector fields

Let M be a closed n(≥2)-dimensional smooth Riemannian manifold and let X be a vector field on M. In this paper, we show that the robust chain transitive set is hyperbolic if and only if there are a C1-neighborhood [Formula: see text] of X and a compact neighborhood U of the chain transitive set such that for any [Formula: see text], the index of the continuation on ΛY(U) = ⋂t∈ℝYt(U) of every critical point does not change.

Bifurcating attractors and Galerkin approximates

Let u̇ = A0u + μA1u + J (u) be a Navier-Stokes parameterized evolution equation in a Hilbert space H and let F1 ⊂ F2 ⊂ F3 ⊂ … be an increasing sequence of finite dimensional spaces such that every Fn ⊕ ℝ contains the center-unstable linear subspace Hu ⊕ ℝ ⊂ H ⊕ ℝ of the system u̇ = A0u + μA1u + J (u), u̇ = 0. Then each Fn ⊕ ℝ determines a Galerkin approximant of the original system, with the same center-unstable linear subspace Hu ⊕ ℝ The flow on the center-unstable manifold of the original system may be identified with a parameterized flow on Hu given by x = f∞ (x,μ). The flow on the center-unstable manifold of the Galerkin approximant determined by Fn ⊕ ℝ may be identified with a parameterized flow on Hu given by ẋ = fn (x,μ). It is proved that Theorem I holds: in the Cktopology on a compact neighborhood of the origin in Hu ⊕ ℝ From this theorem it is concluded that Theorem 2 holds: If a certain priori bound holds relating f∞ and fn and an asymptotically stable set A of ẋ = fn (x,μ) near the origin, then ẋ = f∞ (x,μ) has an asymptotically stable set near the origin with the same Borsuk shape as A. Conversely, for each asymptotically stable set near the origin of ẋ = f∞(x,μ), there is one of the same Borsuk shape for ẋ = fn (x,μ) provided n is large enough. Informally, these results amount to the statement that asymptotically stable sets of the Navier-stokes equation, bifurcating from a steady solution, are recovered up to Borsuk shape by those of large enough Galerkin approximants.