-
Previous Article
Periodic perturbations of scalar second order differential equations
- DCDS Home
- This Issue
-
Next Article
Expansion rates and Lyapunov exponents
Periodic orbits on Riemannian manifolds with convex boundary
1. | Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 BARI, Italy |
$D_s\dot x(s) + \nabla V_x(x(s),s) = 0$
where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.
[1] |
Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097 |
[2] |
Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197 |
[3] |
Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155 |
[4] |
YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure and Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1 |
[5] |
Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020 |
[6] |
Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems and Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135 |
[7] |
David M. A. Stuart. Solitons on pseudo-Riemannian manifolds: stability and motion. Electronic Research Announcements, 2000, 6: 75-89. |
[8] |
Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177 |
[9] |
Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 |
[10] |
Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949 |
[11] |
Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451 |
[12] |
Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967 |
[13] |
Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331 |
[14] |
Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005 |
[15] |
Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403 |
[16] |
Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701 |
[17] |
Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure and Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022 |
[18] |
Alberto Farina, Jesús Ocáriz. Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1929-1937. doi: 10.3934/dcds.2020347 |
[19] |
Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007 |
[20] |
Hyunjin Ahn, Seung-Yeal Ha, Woojoo Shim. Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds. Kinetic and Related Models, 2021, 14 (2) : 323-351. doi: 10.3934/krm.2021007 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]