Attractiveness of Invariant Manifolds

TL;DR

This paper introduces a universal, simple theory for the attractiveness of invariant manifolds in dynamical systems, based on the Lyapunov method, providing criteria for attraction or repulsion.

Contribution

It offers a new, operable criterion for invariant manifold attractiveness using geometric angles and inner products, extending previous Lyapunov-based approaches.

Findings

Invariant manifolds are attractive if the tangent vectors form obtuse angles with normals.

Repulsive manifolds occur when tangent vectors form acute angles with normals.

The theory is supported by examples including equilibria and periodic solutions.

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.