Invariant sub-manifolds and modes of nonlinear autonomous systems

A definition of the modes of a nonlinear autonomous system was developed. T he existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem , center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.