Leading invariant manifold

Observe that near each node point the invariant curve coincides with its leading manifolds. It follows that the invariant curve has a finite smoothness, generally speaking. 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

In summary, the principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. The electromagnetic field is the compensating field of local phase transformation and the gravitational field is the compensating field of local Lorentz transformations. 

In order to make more direct correspondence between tangency and global changes in the dynamical behavior, we propose to use different methods to characterize chaos. The first one focuses attention on how normally hyperbolic invariant manifolds are connected with each other by their stable and unstable manifolds. Then, crisis would lead to a transition in their connections. The second one is to characterize chaos based on how unstable manifolds are folded as they approach normally hyperbolic invariant manifolds. Then, crisis would manifest itself as a change in their folding patterns. Let us explain these ideas in more detail. 

The question of the construction of a nontrivial smooth SC-metric on S was solved in 1902 by Zoll [190] who explicitly constructed a real-analytic SC rotation surface homeomorphic to 5. A simple generalization of the Zoll construction leads to constructing a family of smooth 5C-metrics on S dependent on the functional parameter and invariant under rotations around a certain axis [I9lj. In his book [192] Blaschke proposed an exquisite modification of the Zoll construction which allows us to construct 5C-manifolds which are homeomorphic to the sphere but are not rotation surfaces. Blaschke surfaces are glued of pieces of different SC rotation surfaces by means of films isometric to parts of the standard sphere. 

We chose as examples the formulations described in Table 1. Remark that these formulae apply only in the case where M (and hence A) are constant the generalisation is essentially trivial but leads to bulkier formulae. Formulation 1 is the standard DAE formulation of index 1 solved for the kinematical variables, as described for instance in [12]. Formulations 1 to 4 were already described and discussed in [20], whereas formulation 5 corresponds to the "projected equations of motion" presented in [18,19]. The term (- Xpo) in formulation 6 causes the invariant manifold (pp = 0) to be asymptotically stable. 

Another case studied in [121] corresponds to the bifurcations of a heteroclinic cycle when the saddle values have opposite signs at equilibrium state Oi and O2 (the case where both saddle values are positive leads either to complex dynamics, if 0 and O2 are both saddle-foci, or reduces to the preceding one by a reversal of time and reduction to the invariant manifold). The main assumption here is that both 0 and O2 are simple saddles (not saddle-foci). 

The unstable manifold Wq is composed of the saddle point itself and two trajectories Fi 2 that come from O as t —> +oo. The stable manifold W is two-dimensional. The leading stable direction in Wq is given by the eigenvector corresponding to the smallest negative characteristic root. In our case, this is Ai = —6, and the corresponding eigenvector is (0,0,1). Note that there is an invariant line x = y = 0 in Wq. ... 

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