Singular hyperbolic systems

Abstract

Abstract. We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.

Some properties of singular hyperbolic attractors

“…We need to make several remarks concerning the terminology adopted in this paper. These remarks relate to a hyperbolic flow, which is a special case of a singular hyperbolic flow (a singular hyperbolic flow that has no fixed points is hyperbolic; see [11]). …”

“…Theorem 2.6 (see [11]). Let Λ 1 ∈ U be an invariant closed set that does not contain any fixed points.…”

“…The formṗ 0 + p 0 D is orthogonal to the vector X(x), and so α = 0 in equation (11). Consequently, the form p 0 (x) satisfies the equatioṅ…”

“…If the flow on a three-dimensional Riemannian manifold generated by a vector field X(x) is robustly topologically transitive on a locally maximal set Λ, then Λ is a singular hyperbolic attractor either for the flow generated by the vector field X(x) or for the flow generated by the vector field −X(x). Theorem 1.5 (see [11]). Let Λ be a topologically transitive attractor of a singular hyperbolic flow and suppose that the one-dimensional separatrices of the fixed points are not contained in the stable manifolds of periodic trajectories and fixed points.…”

“…It was shown in [11] that if the manifold M is three-dimensional and the attractor is topologically transitive, then the eigenvalues of the linearized flow in a neighbourhood of a fixed point are such that λ 1 > 0 > λ 2 > λ 3 , λ 1 + λ 2 > 0.…”