Abstract
Let us consider a pair of periodic orbits which degenerate and cancel, through a simple degeneracy. We wish to prove, in this section, that another simple degeneracy, for another type of curves, which are labeled false critical points at infinity of the second kind, takes place at the same time. For sake of simplicity, we will assume that the degenerate periodic orbit has Morse index (including nullity) equal to 2. The Morse index is computed for the functional J(x) = a for x ∈ Cβ, having \(\dot x = a\xi + bv\). Since this index is 2, the H 10 -index at any time t0 (index with a nullity for the w-component η of the tangent vector field z = λξ + μυ + ηw at t0) is at least 1. This implies, because υ turns in any ξ-transported frame, see Proposition A3 of [2], that the degeneracy direction u coincides with the direction ±υ at least once and, in fact, exactly twice. All the claims stated above require, for their complete proofs, technical arguments which are not very difficult and are skipped here. Setting the time zero on the degenerating periodic orbit at a time when ucoincides with y and setting, for sake of simplicity, the time whereucoincides with –v to be 1/2, denoting l/isthe one-parameter group of, we have
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© 2003 Springer Science+Business Media New York
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Bahri, A. (2003). False Critical Points at Infinity of the Second Kind. In: Flow Lines and Algebraic Invariants in Contact Form Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 53. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0021-5_7
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DOI: https://doi.org/10.1007/978-1-4612-0021-5_7
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6576-4
Online ISBN: 978-1-4612-0021-5
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