Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

Previous Article
Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation
DCDS Home
This Issue
Next Article
Euler-Poisson equations related to general compressible rotating fluids

July 2011, 29(3): 1097-1111. doi: 10.3934/dcds.2011.29.1097

Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

Fei Liu ^1, , Jaume Llibre ^2, and Xiang Zhang ^1,

1.	Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China, China
2.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona

Received March 2009 Revised September 2010 Published November 2010

Abstract
Related Papers

Let $\mathcal M$ be a smooth Riemannian manifold with the metric $(g_{ij})$ of dimension $n$, and let $H= 1/2 g^{ij}(q)p_ip_j+V(t,q)$ be a smooth Hamiltonian on $\mathcal M$, where $(g^{ij})$ is the inverse matrix of $(g_{ij})$. Under suitable assumptions we prove the existence of heteroclinic orbits of the induced Hamiltonian systems.

Keywords: Heteroclinic orbit, variational method, Hamiltonian system, Riemannian manifold..

Mathematics Subject Classification: Primary: 37C29, 37J45, 70H05; Secondary: 34C37, 37C1.

Citation: 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

References:

[1]	V. I. Arnold, "Mathematial Methods of Classical Mechanics," 2nd edition, Springer, New York, 1989.
[2]	A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman & Hall/CRC, Boca Raton, FL, 2004.
[3]	K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems," Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2005.
[4]	E. Canalias and J. J. Masdemont, Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems, Discrete Contin. Dyn. Syst., 14 (2006), 261-279.
[5]	P. C. Carriã and O. H. Miyagaki, Existence of homoclinic solutions for a class of time dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184.
[6]	Ch. N. Chen and S. Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Diff. Eqns., 177 (2001), 121-145. doi: 10.1006/jdeq.2000.3996.
[7]	C. Chen, F. Liu and X. Zhang, Orthogonal separable Hamitonian systems on $T^2$, Science in China Ser. A, 50 (2007), 1725-1737. doi: 10.1007/s11425-007-0156-7.
[8]	M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992.
[9]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983.
[10]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.
[11]	M. Izydorek and J. Janczewska, Heteroclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 238 (2007), 381-393. doi: 10.1016/j.jde.2007.03.013.
[12]	J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963.
[13]	J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003.
[14]	P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346.
[15]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
[16]	P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985.
[17]	P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127.
[18]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 472-499. doi: 10.1007/BF02571356.
[19]	W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987.
[20]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.

show all references

References:

[1]	V. I. Arnold, "Mathematial Methods of Classical Mechanics," 2nd edition, Springer, New York, 1989.
[2]	A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman & Hall/CRC, Boca Raton, FL, 2004.
[3]	K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems," Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2005.
[4]	E. Canalias and J. J. Masdemont, Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems, Discrete Contin. Dyn. Syst., 14 (2006), 261-279.
[5]	P. C. Carriã and O. H. Miyagaki, Existence of homoclinic solutions for a class of time dependent Hamiltonian systems, J. Math. Anal. Appl., 230 (1999), 157-172. doi: 10.1006/jmaa.1998.6184.
[6]	Ch. N. Chen and S. Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Diff. Eqns., 177 (2001), 121-145. doi: 10.1006/jdeq.2000.3996.
[7]	C. Chen, F. Liu and X. Zhang, Orthogonal separable Hamitonian systems on $T^2$, Science in China Ser. A, 50 (2007), 1725-1737. doi: 10.1007/s11425-007-0156-7.
[8]	M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992.
[9]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983.
[10]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.
[11]	M. Izydorek and J. Janczewska, Heteroclinic solutions for a class of the second order Hamiltonian systems, J. Diff. Eqns., 238 (2007), 381-393. doi: 10.1016/j.jde.2007.03.013.
[12]	J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963.
[13]	J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003.
[14]	P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346.
[15]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
[16]	P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985.
[17]	P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127.
[18]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 472-499. doi: 10.1007/BF02571356.
[19]	W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987.
[20]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.

[1]	Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[2]	Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[3]	Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20.
[4]	Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks and Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021
[5]	Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015
[6]	Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[7]	Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799
[8]	Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173
[9]	Andrea Venturelli. A Variational proof of the existence of Von Schubart's orbit. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 699-717. doi: 10.3934/dcdsb.2008.10.699
[10]	Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769
[11]	Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[12]	Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507
[13]	Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[14]	Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019
[15]	Ella Pavlechko, Teemu Saksala. Uniqueness of the partial travel time representation of a compact Riemannian manifold with strictly convex boundary. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022028
[16]	Paula Balseiro, Teresinha J. Stuchi, Alejandro Cabrera, Jair Koiller. About simple variational splines from the Hamiltonian viewpoint. Journal of Geometric Mechanics, 2017, 9 (3) : 257-290. doi: 10.3934/jgm.2017011
[17]	Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024
[18]	Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039
[19]	Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[20]	Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Fei Liu Jaume Llibre Xiang Zhang
on Google Scholar
Fei Liu Jaume Llibre Xiang Zhang

[Back to Top]