The third one is Melnikov's criterion which is a powerful approximate tool for investigating chaos occurrence in near Hamiltonian systems and has been successfully applied to the analysis of chaos in smooth systems by calculating the distance between the stable and

unstable manifold [18].

Then there exist a stable manifold [W.sup.s] tangent to [E.sup.s], an

unstable manifold Ws tangent to [E.sup.u], and a centre manifold [W.sup.c] tangent to [E.sup.c] at [x.sub.c].

(a) If [[alpha].sub.2] > [delta] then [[lambda].sub.1] > [delta] and [[lambda].sub.2] > 0; then [U.sup.*.sub.0] is an unstable node and its

unstable manifold is [E.sup.u] = <(1; 0), (0; 1)> = [R.sup.2].

As the manifold M along with its stable manifold [W.sup.s] (M) and

unstable manifold [W.sup.u](M) is invariant under sufficiently small perturbations [24], under perturbation (when [epsilon] [not equal to] 0),

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] mean that [[GAMMA].sup.1] is a heteroclinic orbit with orbit flip, [W.sup.cu.sub.i] is the center

unstable manifold of [p.sub.1], [W.sup.u.sub.i] (resp., [W.sup.S.sub.i]) is the unstable (resp., stable) manifold of [p.sub.i], and [W.sup.uu.sub.i] (resp., [W.sup.ss.sub.i]) is the strong unstable (resp., stable) manifold of [p.sub.i], i = 1,2.

In Section 3.1.3, the behavior of trajectories on the strongly

unstable manifold at ((r/a), 0, 0) is presented by some technical lemmas.

All spacecrafts around the hyperbolic eccentricity cannot be maintained on such an orbit; on the contrary, they will move on the

unstable manifold towards higher eccentricities, or on the

unstable manifold towards lower eccentricities based on the quite small change of the orbital elements.

This also tells us the

unstable manifold of the origin is one-dimensional.

Denote by [[LAMBDA].sub[[gamma].sub.b]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a closure of the

unstable manifold of [[LAMBDA].sub[[gamma].sub.b]] at [P.sub.+]([[LAMBDA].sub[[gamma].sub.b]]), a neighbourhood of [P.sb.+]([[LAMBDA].sub[[gamma].sub.b]]) and a neighbourhood of [LAMBDA].sub[[gamma].sub.b]]([LAMBDA].sub[[gamma].sub.b]]), respectively.

The circular area continuously deforms along the (perturbed)

unstable manifold and tangles when it approaches to the (perturbed) hyperbolic fixed point.

The

Unstable Manifold: Janet Frame's Challenge to Determinism.

and the [epsilon]-local

unstable manifold through z by: