Attractiveness of Invariant Manifolds of Two Dimensional Dynamical Systems

In this paper an operable, universal and simple theory on the attractiveness of the invariant manifolds of the two-dimensional dynamical systems is first obtained. It is motivated by the Lyapunovdirect method. It means that for any point x^→ in the invariant manifold M, n(x^→) is the normal passing by x^→, and ∀x^→ ∈n(x^→), if the tangent f(x^→) of the orbit of the dynamical system intersects at obtuse (sharp) angle with the n(x^→), or the inner product of the normal vector n^→(x^→) and tangent vector f^→(x^→) is negative (positive), i.e., f^→(x^→). n^→(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 this result.