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SCIENTIA SINICA Mathematica, Volume 46 , Issue 5 : 741-746(2016) https://doi.org/10.1360/N012015-00379

Multiplicity problems of brake orbits with prescribed energy in Hamiltonian systems

More info
  • ReceivedNov 30, 2015
  • AcceptedJan 7, 2016
  • PublishedApr 26, 2016

Abstract


Funded by

国家自然科学基金(11271200)

国家自然科学基金(11422103)


References

[1] Long Y. Index Theory for Symplectic Paths with Applications. Basel: Birkhäuser, 2002. Google Scholar

[2] Seifert H. Periodische bewegungen mechanischer systeme. Math Z, 1948, 51: 197-216. Google Scholar

[3] Bolotin S. Libration motions of natural dynamical systems (in Russian). Vestnik Moskov Univ Ser I Mat Mekh, 1978, 6: 72-77. Google Scholar

[4] Hayashi K. Periodic solution of classical Hamiltonian systems. Tokyo J Math, 1983, 6: 473-486. Google Scholar

[5] Gluck H, Ziller W. Existence of periodic solutions of conservative systems. In: Seminar on Minimal Submanifolds. Princeton: Princeton University Press, 1983, 65-98. Google Scholar

[6] Benci V. Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann Inst H Poincar$\acute{\rm {e}}$ Anal Non Lin$\acute{\rm {e}}$aire, 1984, 1: 401-412. Google Scholar

[7] Rabinowitz P H. On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear Anal, 1987, 11: 599-611. Google Scholar

[8] Weinstein A. Normal modes for nonlinear Hamiltonian systems. Invent Math, 1073, 20: 47-57. Google Scholar

[9] Giamb${\rm \grave{o}}$ R, Giannoni F, Piccione F. Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds. Adv Differential Equations, 2005, 10: 931-960. Google Scholar

[10] Giamb${\rm \grave{o}}$ R, Giannoni F, Piccione F. Multiple brake orbits in $m$-dimensional disks. Calc Var Partial Differential Equations, 2015, 54: 2553-2580. Google Scholar

[11] Szulkin A. An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems. Math Ann, 1989, 283: 241-255. Google Scholar

[12] van Groesen E W C. Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy. J Math Anal Appl, 1988, 132: 1-12. Google Scholar

[13] Ambrosetti A, Benci V, Long Y. A note on the existence of multiple brake orbits. Nonlinear Anal, 1993, 21: 643-649. Google Scholar

[14] Long Y, Zhang D, Zhu C. Multiple brake orbits in bounded convex symmetric domains. Adv Math, 2006, 203: 568-635. Google Scholar

[15] Liu C, Zhang D. Iteration theory of $L$-index and multiplicity of brake orbits. J Differential Equations, 2014, 257: 1194-1245. Google Scholar

[16] Zhang D, Liu C. Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $\R^{2n}$. Ann Inst H Poincar$\acute{\rm {e}}$ Anal Non Lin$\acute{\rm {e}}$aire, 2014, 31: 531-554. Google Scholar

[17] Zhang D, Liu C. Multiplicity of brake orbits on compact convex symmetric reversible hypersurfaces in $\R^{2n}$ for $n\ge 4$. Proc Lond Math Soc (3), 2013, 107: 1-38. Google Scholar

[18] Liu C, Zhang D. Seifert conjecture in the even convex case. Comm Pure Appl Math, 2014, 67: 1563-1604. Google Scholar

[19] Cappell S E, Lee R, Miller E Y. On the Maslov-type index. Comm Pure Appl Math, 1994, 47: 121-186. Google Scholar

[20] Long Y, Zhu C. Closed characteristics on compact convex hypersurfaces in $\R^{2n}$. Ann Math (2), 2002, 155: 317-368. Google Scholar

[21] Liu C. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete Contin Dyn Syst, 2010, 27: 337-355. Google Scholar

[22] Zhang D. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete Contin Dyn Syst, 2015, 35: 2227-2272. Google Scholar

[23] Dell'Antonio G, D'Onofrio B, Ekeland I. Les systémes hamiltoniens convexes et pairs ne sont pas ergodiques en général. C R Acad Sci Paris Sér I Math, 1992, 315: 1413-1415. Google Scholar

[24] Wang W. Stability of closed characteristics on symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. J Math Pures Appl (9), 2013, 99: 297-308. Google Scholar

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