Minimal P-symmetric period problem of first-order autonomous Hamiltonian systems

Ambrosetti A, Coti Zelati V. Solutions with minimal period for Hamiltonian systems in a potential well. Ann Inst H Poincaré Anal Non Linéaire, 1987, 4: 242–275 Ambrosetti A, Mancini G. Solutions of minimal period for a class of convex Hamiltonian systems. Math Ann, 1981, 255: 405–421 Chenciner A, Montgomery R. A remarkable periodic solution of the three body problem in the case of equal masses. Ann of Math, 2000, 152(3): 881–901 Clarke F, Ekeland I. Hamiltonian trajectories having prescribed minimal period. Comm Pure Appl Math, 1980, 33(3): 103–116 Dong D, Long Y. The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems. Trans Amer Math Soc, 1997, 349: 2619–2661 Dong Y. P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems. Nonlinearity, 2006, 19(6): 1275–1294 Dong Y, Long Y. Closed characteristics on partially symmetric compact convex hypersurfaces in ℝ2n: J Differential Equations, 2004, 196: 226–248 Ekeland I, Hofer H. Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems. Invent Math, 1985, 81: 155–188 Fei G. Relative Morse index and its application to Hamiltonian systems in the presence of symmetries. J Differential Equations, 1995, 122: 302–315 Fei G, Kim S K, Wang T. Minimal period estimates of period solutions for superquadratic Hamiltonian systems. J Math Anal Appl, 1999, 238: 216–233 Fei G, Qiu Q. Minimal period solutions of nonlinear Hamiltonian systems. Nonlinear Anal, 1996, 27(7): 821–839 Ferrario D, Terracini S. On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent Math, 2004, 155(2): 305–362 Hu X, Sun S. Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit. Comm Math Phys, 2009, 290: 737–777 Hu X, Sun S. Stability of relative equilibria and Morse index of central configurations. C R Math Acad Sci Paris, 2009, 347: 1309–1312 Hu X, Sun S. Morse index and the stability of closed geodesics. Sci China Math, 2010, 53(5): 1207–1212 Hu X, Wang P. Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems. J Funct Anal, 2011, 261: 3247–3278 Liu C. Maslov P-index theory for a symplectic path with applications. Chin Ann Math Ser B, 2006, 27(4): 441–458 Liu C. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete Contin Dyn Syst, 2010, 27(1): 337–355 Liu C. Periodic solutions of asymptotically linear delay differential systems via Hamiltonian systems. J Differential Equations, 2012, 252: 5712–5734 Liu C. Relative index theories and applications. Topol Methods Nonlinear Anal (to appear) Liu C, Long Y. An optimal increasing estimate of the iterated Maslov-type indices. Chinese Sci Bull, 1997, 42: 2275–2277 Liu C, Long Y. Iteration inequalities of the Maslov-type index theory with applications. J Differential Equations, 2000, 165: 355–376 Liu C, Tang S. Maslov (P, ω)-index theory for symplectic paths. Adv Nonlinear Stud, 2015, 15: 963–990 Liu C, Tang S. Iteration inequalities of the Maslov P-index theory with applications. Nonlinear Anal, 2015, 127: 215–234 Long Y. The minimal period problem for classical Hamiltonian systems with even potentials. Ann Inst H Poincaré Anal Non Linéaire, 1993, 10: 605–626 Long Y. The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems. J Differential Equations, 1994, 111: 147–174 Long Y. On the minimal period for periodic solutions of nonlinear Hamiltonian systems. Chin Ann Math Ser B, 1997, 18: 481–484 Long Y. Index Theory for Symplectic Paths with Application. Progr Math, Vol 207. Basel: Birkhäuser Verlag, 2002 Long Y, Zhu C. Closed characteristics on compact convex hypersurfaces in ℝ2n: Ann of Math, 2002, 155: 317–368 Rabinowitz P H. Periodic solutions of Hamiltonian systmes. Comm Pure Appl Math, 1978, 31: 157–184