://dx.doi.org/10.1515/9783110858372  Field M.J., Dynamics and Symmetry, ICP Adv. Texts Math., 3, Imperial College Press, London, 2007  Floer A., A refinement of the Conley index and an application to the stability of hyperbolicinvariantsets, Ergodic Theory Dynam. Systems, 1987, 7(1), 93–103 http://dx.doi.org/10.1017/S0143385700003825  Floer A., Zehnder E., The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergodic Theory Dynam. Systems, 1988, 8, 87–97 http://dx.doi.org/10.1017/S0143385700009354  Gęba K., Degree for gradient
capture transfer used by Hiten arriving at Moon
on October 2, 1991. 147
3.16 Nominal exterior WSB-transfer. 150
3.17 Capture process. 154
3.18 Time sections
t to periodic orbit ψ having a transverse
homoclinic orbit φ. A map σ is defined on
t with trans-
verse homoclinic point r. The sections are all identified. 170
3.19 Intersection of Wu(0),Ws(0) on Q3 = 0. 173
3.20 Region near infinity in McGehee coordinates. 174
3.21 Map ψ near infinity. 175
3.22 Construction of the transversal map, φ̃. 177
3.23 Kepler flow in McGehee coordinates. 181
This chapter comprises about one-half of this book, and a number of different
topics are covered pertaining to the subject of capture in the three- and four-
body problems. Two forms of capture are studied, and in the last section we
prove that they are equivalent on a hyperbolicinvariantset for the circular
Section 3.1 serves as an introduction to the different types of capture and
other dynamics we will consider in this chapter. These include permanent
capture and a theorem by Chazy and Hopf on the measure of
. Floer, A refinement of the Conley index and application to the stability of hyperbolicinvariantsets,
Erg. Th. and Dyn. Sys. 7 (1987), 93-103,
 K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Birkhäuser, Topological
Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu i A. Vignoli, Progr. in Nonl.
Diff. Equat. and Their Appl. 27, Birkhäuser, (1997), 247-272,
 A. Gołȩbiewska and S. Rybicki, Bifurcations of critical orbits of invariant strongly indefinite functionals,
accepted for publication to Nonl. Anal. TMA
system (2.3) connecting normally hyperbolicinvariantsets implies the existence of
a transversal homoclinic orbit (transversal heteroclinic orbit) of system (2.1) for
sufficiently small e.
Now we consider the case of a non-transversal homoclinic orbit. To this end
(A3) For λ = λο, the degenerate system (2.3) has a hyperbolic equilibrium point
u0 , that is, the Jacobian gu(uo,^o) has no eigenvalues on the imaginary
(A4) For λ = λ0 , the degenerate system (2.3) has a homoclinic orbit 7 to u0, that
is, there is a solution 7 : R —> Rn of
chaotic process by proving the existence of a
There are several new theorems and results in this book that have not
been previously published. These include Theorem 2.33 on the existence
of Aubrey-Mather sets in the restricted problem, and Theorem 3.58 on the
proof of existence of an invariant hyperbolic network on an extended version
of the weak stability boundary. A short proof is given for Theorem 1.39 on
the existence of geodesic flows for the Kepler problem.
Scope of This Book
The approach taken here is mathematical and is both
a specific set.
1.0.2 Limit of the first eigenvalue and the topological pressure
Under hyperbolicity assumptions on the field b, the limit of the sequence " has
been determined in [10, 31]. It was based on the fact that " is given by the
following variational formula (see ),
" D sup
where belongs to P.V /, the space of probability measure on V . It is proved
in  that when the recurrent set K of the field b is a finite union of isolated
components K1; K2; : : : ; KN which are hyperbolicinvariantsets, the limit
the transverse directions.
A classical example of a strange attractor is the Lorenz attractor. There
are many other examples of invariant sets with “wild” topological structure.
Among them are the well-known Smale-Williams solenoid and Smale horseshoe.
The latter is an example of a hyperbolicinvariantset whose local topological
structure is the product of different Cantor-like sets. Strange attractors are
always associated with trajectories having extremely irregular behavior and are
thought of as the origin of “dynamical” chaos.
Specialists in dynamical