Omega-limit sets close to singular-hyperbolic attractors

Details Omega-limit sets close to singular-hyperbolic attractors Omega-limit sets close to singular-hyperbolic attractors

2003-07-23

arxiv.org/abs/math/0307316v1

Mathematics

Dynamical Systems

17 pages

Scientific paper

We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $ \{x\in U:\omega_Y(x)$ contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the persistence of singular-hyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor \cite{gw} and the example in \cite{mpu}.

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