b) Melnikov method is used in case of periodic motions and is analysing the conditions necessary in order for the stable and non-stable varieties of the same hyperbolic point to transversally intersect each odder.
If [[??].sub.0] < [[??].sub.0cr], the stable and non-stable varieties of the hyperbolic point cannot have a transversal intersection and, as a general conclusion, the chaotically motions may not occur.
The realization of each vertex u, u [member of] V (M) in [R.sup.3] space is shown in the Fig.1 for each case of [rho](u)[mu](u) > 2[pi],= 2[pi] or < 2[pi], call elliptic point, euclidean point and hyperbolic point, respectively.
Here, we assume the angle at the intersection point is in clockwise, that is, a line passing through an elliptic point will bend up and a hyperbolic point will bend down, such as the cases (b),(c) in the Fig.2.