Invariant foliation

The key methods in our presentation of local bifurcations are based on the center manifold theorem and on the invariant foliation technique (see Sec. 5.1. of Part I). The assumption that there are no characteristic exponents to the right of the imaginary axis (or no multipliers outside the unit circle) allows us to conduct a smooth reduction of the system to a very convenient standard form. We use this reduction throughout this book both in the study of local bifurcations on the stability boundaries themselves and in the study of global bifurcations on the route over the stability boundaries (Chap. 12).These... 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

In a neighborhood of the point O there exists a -smooth change of variables which straightens both the invariant foliation and the center manifold so that the system in the new variables assumes the following standard form... 

Here, the center manifold is defined by the equation y = 0. The surfaces x = constant are the leaves of the strong-stable invariant foliation In particular, x = 0 is the equation of the strong-stable manifold of O. At fi — Oj the function g (nonlinear part of the map on W ) has a strict extremum at X = 0. For more definiteness, we assume that it is a minimum, i.e. y(x, 0) > 0 when X 0. Thus, the saddle region on the cross-section corresponds to x > 0, and the node region corresponds to x < 0. Since the saddle-node disappears when /Lt > 0, it follows that y(x,/x) > 0 for all sufficiently small x and for all small positive //. 

Next, let us straighten the strong stable invariant foliation. The leaves of the foliation are given by x Q y], x p), (p = constant where x is the coordinate of intersection of a leaf with the center manifold Q is a C -function (it is C -smooth with respect to y). The straightening is achieved via a coordinate transformation Xh- which brings the invariant foliation to the form x = constant,

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Already we have stated that the invariant foliation is C -smooth moreover, it can be shown for the case of the saddle-node that the foliation is, in fact, C -smooth everywhere except on Wff at p 0 [140]. The coordinate transformation that reduces (12.2.4) to (12.2.7) has the same smoothness. 

This is the uniquely defined invariant foliation of transverse to the leading direction near O see Sec. 6.1. 

Since 4> does not appear explicitly in the Hamiltonian, we go one step further, exploiting the other constant of the motion, A (rotational invariance of the Hamiltonian). Let us define a torus T2 C S3 in the following way. Since A is a conserved quantity, the A = Aq surfaces foliate the S3 (Pi ) sphere in a... 

In Fig. 6, the dynamics on can be decomposed into the movement along the normal directions and the flow on the manifold Mg. Note that the time development of b does not affect the movement of the base points. Suppose two points PI and PI on with the same base point yl. Then, the orbit from PI through P2 reaching P3 and the one from PI through P2 reaching P3 are projected to the same movement of the base points on from yl through y2 reaching y3. In other words, the three-dimensional invariant manifold Wl consists of two-dimensional invariant manifolds that correspond to the movements of the base points. Then, in mathematics, we say that the three-dimensional stable manifold is foliated by two-dimensional leaves. This structure is called foliation [28,30]. 

According to Theorem C.6, the limit set can be deformed to a compact invariant set A, without rest points, of a planar vector field. By the Poin-care-Bendixson theorem, A must contain at least one periodic orbit and possibly entire orbits which have as their alpha and omega limits sets distinct periodic orbits belonging to A. Using the fact that A is chain-recurrent, Hirsch [Hil] shows that these latter orbits cannot exist. Since A is connected it must consist entirely of periodic orbits that is, it must be an annulus foliated by closed orbits. Monotonicity is used to show... 

According to Nekhoroshev (1977) and to Morbidelli and Giorgilli (1995), the old and crucial question of stability of a dynamical system turns out to be related to the structure and density of invariant tori which foliate the phase space. For instance the puzzle of the 2/1 gap of the asteroidal belt distribution was explained showing that the corresponding region of the phase space is a weak chaotic one (Nesvorny and Ferraz-Mello 1997). 

The NHIM has a special structure due to the conservation of the center actions, it is filled, or foliated, by invariant d — l)-dimensionaI tori. More precisely, for d = 3 DoFs, each value of Jz implicitly defines a value of h by the energy equation Kcnf(0,/2,/3) = E. For three DoFs, the NHIM is thus foliated by a one-parameter family of invariant 2-tori. The end points of the parameterization interval correspond to Jz = 0 (implying qz = Pi = 0) and /s = 0 (implying q3=pz = 0), respectively. At the end points, the 2-tori thus degenerate to periodic orbits, the so-called Lyapunov periodic orbits. 

Fig. 9.3.3. When all Lyapunov values vanish in an anzilytical system, the equilibrium state is a center on W. In its extended neighborhood is foliated by invariant cylinders.

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