Attractiveness of Invariant Manifolds

Abstract

In this paper an operable, universal and simple theory on the attractiveness of the invariant manifolds is first obtained. It is motivated by the Lyapunov direct method. It means that for any point $\overrightarrow{x}$ in the invariant manifold $M$, $n(\overrightarrow{x})$ is the normal passing by $\overrightarrow{x}$, and $\forall \overrightarrow{x^{'}} \in n(\overrightarrow{x})$, if the tangent $f(\overrightarrow{x^{'}})$ of the orbits of the dynamical system intersects at obtuse (sharp) angle with the normal $n(\overrightarrow{x})$, or the inner product of the normal vector $\overrightarrow{n}(\overrightarrow{x})$ and tangent vector $\overrightarrow{f}(\overrightarrow{x^{'}})$ is negative (positive), i.e., $\overrightarrow{f}(\overrightarrow{x^{'}}). \overrightarrow{n}(\overrightarrow{x}) < (>)0$, then the invariant manifold $M$ is attractive (repulsive). Some illustrative examples of the invariant manifolds, such as equilibria, periodic solution, stable and unstable manifolds, other invariant manifold are presented to support our result.