Index theory for heteroclinic orbits of Hamiltonian systems

Abstract

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits.Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrised by bounded intervals.

The Maslov index and the spectral flow—revisited

“…Our final result generalises Theorem 2 of [9], where the following spectral flow formula was shown for a particular class of functions α, β that satisfy (25). sf(A) = µ Mas (Ψ 0 (α(·))γ 1 (0), Ψ 0 (β(·))Ψ 0 (1) −1 γ 2 (0)) + µ Mas (γ 1 , γ 2 ) − µ Mas (Ψ 1 (α(·))γ 1 (1), Ψ 1 (β(·))Ψ 1 (1) −1 γ 2 (1)).…”

“…. , N , for A as in (9). Let N T,εi be an open neighbourhood of some T ∈ CF sa (H) as in the construction of the spectral flow such that A λ ∈ N T,εi for all λ ∈ [λ i−1 , λ i ].…”

“…Throughout the paper, we aim our presentation to be rather self-contained, and we will just use some well-known facts from Kato [10]. Finally, we review a recent spectral flow formula for linear Hamiltonian systems by Hu and Portaluri from [9], which they call a new index theory on bounded domains. Firstly, we note that the considered families of Hamiltonian systems are continuous with respect to the gap-metric, which follows easily from our approach to Cappell, Lee and Miller's Theorem.…”

Decomposition of spectral flow and Bott-type iteration formula

“…Please see Section 4 for the detail. It is well known that spectral flow is equal to Maslov index, and this is also true for the unbounded domain, see [6,27,26,7,15] and reference therein. The Maslov index is associated integer to a pair of continuous path f (t) = (L 1 (t), L 2 (t)), t ∈ I, in Lag(2n) × Lag(2n) [6].…”

“…In the case of homoclinics, let A λ = A − B * − λ(B − B * ), then from [7] or [15] the index satisfied…”