Stable/unstable manifolds Melnikov integral

To investigate this behavior, we have two methods available at present One is the Melnikov integral, which is used by Arnold in Ref. 15 to see if the stable and unstable manifolds of whiskered tori intersect. The other is Lie perturbation theory, which is used in astrophysics to control orbiters in the universe [22]. 

In this section, we will introduce the Melnikov integral, which estimates the gaps between stable and unstable manifolds. Thus, if the Melnikov integral attains the value zero, this signals the existence of intersections between stable and unstable manifolds. If the intersection is transversal, it implies the existence of horseshoe dynamics [32], that is, chaotic behavior. On the other hand, if the intersection is tangent [11], it implies that the system is at a transition between different kinds of dynamics [12]. Such transitions of chaotic behavior are called crisis [13]. The tangency will be further analyzed in Section VII. 

These assumptions simplify the derivation of the Melnikov integral. Otherwise, the NHIM would shift from Mq, and some tori on would be destroyed due to perturbations, as discussed in Section IV. Then, a discussion which is similar to the KAM theorem says that nonresonant tori remain on the NHIM Me [31]. Using the stable and unstable manifolds of these tori, the derivation of the Melnikov integral follows [24]. 

One of the remarkable aspects of Eq. (51) is that its right-hand side is given only using the unperturbed orbit (jCo(t,7o),7o, 0o(O)- Thus, in obtaining the quantity d (t), we do not need any information on perturbed orbits. This is one of the merits of using the Melnikov integral to estimate the gap between the stable and unstable manifolds. 

Equation (64) shows that the distance d x,a) exhibits an oscillatory dependence as a function of x. In other words, d x, a) changes between plus and minus values as initial conditions shift on the separatrix. This means that the stable and unstable manifolds have transverse intersections. See Fig. 14 showing how the oscillatory change of the integral implies the occurrence of transverse intersections. The existence of transverse intersections between stable and unstable manifolds leads to horseshoe dynamics—that is, chaos. Thus, the Melnikov integral given by Eq. (64) indicates that this system exhibits chaotic behavior. 

Chaos in systems with N degrees of freedom with N >3 has characteristics that are not shared by chaos in systems with two degrees of freedom. In this section, we show that the Melnikov integral reveals one of these characteristics. They are exhibited in the intersections between the stable and unstable manifolds of whiskered tori with different action values. 

Then, how these manifolds intersect will be investigated using the Melnikov integral. See Fig. 17 for a schematic picture of the stable and unstable manifolds with a shift in the variable 7. 

On M-y there are locally stable and unstable manifolds that are of equal dimensions and are close to the impertm-bed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold intersects each of the 5D level energy sm-faces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem. The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. 

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