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. parametrized 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 parametrized by bounded intervals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Actually \(\lambda \mapsto A_\lambda \in \mathscr {BF}^{sa}(\mathcal W, \mathcal V)\).
References
Abbondandolo, A., Majer, P.: Ordinary differential operators in Hilbert spaces and Fredholm pairs. Math. Z. 243(3), 525–562 (2003)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Math. Proc. Camb. Philos. Soc. 79(1), 71–99 (1976)
Abbondandolo, A., Portaluri, A., Schwarz, M.: The homology of path spaces and Floer homology with conormal boundary conditions. J. Fixed Point Theory Appl. 4(2), 263–293 (2008)
Arnol’d, V. I.: On a characteristic class entering into conditions of quantization. (Russian) Funkcional. Anal. i Priloz̆en. 1:1–14 (1967)
Arnol’d, V.I.: Sturm theorems and symplectic geometry. (Russian) Funktsional. Anal. i Prilozhen. 19 (4): 1–10, 95 (1985)
Barutello, V., Jadanza, R.D., Portaluri, A.: Linear instability of relative equilibria for n-body problems in the plane. J. Differ. Equ. 257(6), 1773–1813 (2014)
Barutello, V., Jadanza, R.D., Portaluri, A.: Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory. Arch. Ration. Mech. Anal. 219(1), 387–444 (2016)
Barutello, V., Hu, X., Portaluri, A. Terracini, S.: An index theory for asymptotic motions under singular potentials. Preprint available on https://arxiv.org/pdf/1705.01291.pdf
Booss-Bavnbek, B., Lesch, M., Phillips, J.: Unbounded Fredholm operators and spectral flow. Can. J. Math. 57(2), 225–250 (2005)
Cappell, S.E., Lee, R., Miller, E.Y.: On the Maslov index. Commun. Pure Appl. Math 47(2), 121–186 (1994)
Chen, C.-N., Hu, X.: Maslov index for homoclinic orbits of Hamiltonian systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(4), 589–603 (2007)
Fitzpatrick, P.M., Pejsachowicz, J., Recht, L.: Spectral flow and bifurcation of critical points of strongly-indefinite functionals. I. Gen. Theory. J. Funct. Anal. 162(1), 52–95 (1999)
Fitzpatrick, P.M., Pejsachowicz, J., Stuart, C.A.: Spectral flow for paths of unbounded operators and bifurcation of critical points. Preprint (2006)
Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28(3), 513–547 (1988)
Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math 41(6), 775–813 (1988)
Floer, A.: A relative Morse index for the symplectic action. Commun. Pure Appl. Math 41(4), 393–407 (1988)
Giambò, R., Piccione, P., Portaluri, A.: Computation of the Maslov index and the spectral flow via partial signatures C. R. Math. Acad. Sci. Paris 338(5), 397–402 (2004)
Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. I. Operator Theory: Advances and Applications, 49. Birkhäuser Verlag, Basel, pp. xiv+468 (1990)
Hu, X., Sun, S.: Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit. Commun. Math. Phys. 290, 737–777 (2009)
Hu, X., Sun, S.: Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem. Adv. Math. 223(1), 98–119 (2010)
Hu, X., Long, Y., Sun, S.: Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory. Arch. Ration. Mech. Anal. 213(3), 993–1045 (2014)
Hu, X., Ou, Y.: Collision index and stability of elliptic relative equilibrium in planar \(n\)-body problem. Commun. Math. Phys 348(3), 803–845 (2016)
Long, Y.: Index theory for symplectic paths with applications. Progress in Mathematics, vol. 207. Birkhäuser Verlag, Basel (2002)
Long, Y., Zhu, C.: Maslov-type index theory for symplectic paths and spectral flow (II). Chin. Ann. Math. Ser. B 21(1), 89–108 (2000)
Maslov, V.P.: Asymptotic methods and perturbation theory M.G.U., Moscow (1965)
Musso, M., Pejsachowicz, J., Portaluri, A.: A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds. Topol. Methods Nonlinear Anal. 25(1), 69–99 (2005)
Musso, M., Pejsachowicz, J., Portaluri, A.: Morse index and bifurcation of \(p\)-geodesics on semi Riemannian manifolds. ESAIM Control Optim. Calc. Var. 13(3), 598–621 (2007)
Phillips, J.: Self-adjoint fredholm operators and spectral flow. Can. Math. Bull 39(4), 460–467 (1996)
Piccione, P., Portaluri, A., Tausk, D.V.: Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics. Ann. Global Anal. Geom 25(2), 121–149 (2004)
Rabier, P.J., Stuart, C.A.: Boundary value problems for first order systems on the half-line. Topol. Methods Nonlinear Anal 25(1), 101–133 (2005)
Robbin, J., Salamon, D.: Maslov index for paths. Topology 32(4), 827–844 (1993)
Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. London Math. Soc. 27(1), 1–33 (1995)
Waterstraat, N.: Spectral flow, crossing forms and homoclinics of Hamiltonian systems. Proc. Lond. Math. Soc. (3) 111(2), 275–304 (2015)
Yoshida, T.: Floer homology and splittings of manifolds. Ann. Math. (2) 134(2), 277–323 (1991)
Zhu, C., Long, Y.: Maslov-type index theory for symplectic paths and spectral flow (I). Chin. Ann. Math. Ser. B 20(4), 413–424 (1999)
Acknowledgements
We thank the anonymous referee for fixing some typos and for the comments that improved the presentation of the paper. We are grateful to proff. Nils Waterstraat, Jacobo Pejsachowicz and Chaofeng Zhu, for many stimulating discussions on this subject.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
Xijun Hu is partially supported by NSFC (No.11425105) and NCET.
Alessandro Portaluri is partially supported by the project ERC Advanced Grant 2013 No. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”, by Prin 2015 “Variational methods, with applications to problems in mathematical physics and geometry” No. 2015KB9WPT_001 and by Ricerca locale 2015 “Semi-classical trace formulas and their application in physical chemistry” No. Borr_Rilo_16_01.
