Extended unstable manifold

The manifold contains all trajectories staying in a small neighborhood J7 of r for all negative times. Near O, the manifold coincides with an extended unstable manifold (not unique). Thus, if we denote li p) =... 

In dimensions higher than three, the condition A 0 is an essential nondegeneracy condition. It is important that we use the coordinates in which the system has locally the form (13.4.10) and that the identities (13.4.11) are hold. In these coordinates the intersection of with Sq is the straightline yo = 0, and the intersection of the extended unstable manifold with Si is tangent to the space m = 0 (the extended imstable manifold is a smooth invariant manifold which is transverse to at O). Thus, one can see from (13.4.14) that the condition A 7 0 is equivalent to the condition of transversality of n5i) to at the point M+, i.e. to the transversality condition... 

The latter condition can be interpreted as the transversality of the two-dimensional extended unstable manifold to the stable manifold along... 

In particular, let the dimension of the unstable manifold of Oi be equal to the dimension of the imstable manifold of O2. Besides, let both the stable and unstable leading characteristic exponents at both 0 and O2 be real. Assume also that both heteroclinic orbits F 1 2 enter and leave the saddles along the leading directions. We also assume that the extended unstable manifold of one saddle is transverse to the stable manifold of the other saddle along every orbit Fi 2, and that the extended stable manifold of one saddle is transverse to the unstable manifold of the other saddle along F 1,2 as well. Under... 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

When df is a Kahler manifold, the stable and the unstable manifolds can be expressed purely in terms of the group action. Notice that an T-action on X extends uniquely to a holomorphic T -action on X. 

Figure 11. Schematic view of a TS (thick black line), with the same type of view as in Fig. 10. The equilibrium point is in the middle with its stable manifold and unstable manifold extending as straight lines. Trajectories in dot-dashed lines are reactive (inside the tubes) and cross TS trajectories in dashed lines are not reactive. The whole gray surface is the energy level. For a linear motion, it takes the form of a parabolic hyperboloid.

This reasoning is not limited to homoclinic intersections. This argument can be easily extended to include heteroclinic intersections where stable and unstable manifolds may have different dimensions. 

In order to proceed further, we need a method to compute stable and unstable manifolds, and see if they intersect. We are currently studying this problem by extending the Lie perturbation methods. The results will be published elsewhere [30]. 

Note also that these results can be extended immediately to the case of a saddle with a multi-dimensional unstable manifold. Namely, if O has several characteristic exponents with positive real parts but the leading characteristic exponent 71 is real, i.e. if... 

Note that all of these results (except for the subtle structure of the set of curves C12 in the case where Oi is a saddle-focus and O2 is a saddle) are proven for C -smooth systems. Therefore, just like in the case of a homoclinic-8, these results can be directly extended to the case where the unstable manifolds of Oi and O2 are multi-dimensional (but they must have equal dimensions in this case), provided that the conditions of Theorem 6.4 in Part I of this book, which guarantee the existence of an invariant -manifold near the heteroclinic cycle, are satisfied. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

De Leon and co-workers [34—37] established an elegant reaction theory for a system with two DOFs, the so-called reactive island theory to mediate reactions through cylindrical manifolds apart from the saddles. Their original algorithm depends crucially on the existence of pure unstable periodic orbits in the nonreactive DOFs in the region of the saddles and did not extend to systems with many DOFs. 

It must be underlined that the central manifold theorem, extending the linear center manifold into the nonlinear regime, is way less powerful than its stable/ unstable counterpart. There is no limit t —> oo and even no unicity of nonlinear center manifolds. Consequently, it is not well known how this whole beautiful stmcture bifurcates and disappears as E > E. There has been virtually no study of the bifurcation stmcture (see, however, Ref. 55), and the transition from threshold behavior to far-above-threshold behavior is an open question, as far as I am aware. 

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