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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

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 & 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.  Google Scholar

[2]

A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman $&$ Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992.  Google Scholar

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D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963.  Google Scholar

[13]

J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003.  Google Scholar

[14]

P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346.  Google Scholar

[15]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  Google Scholar

[16]

P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985.  Google Scholar

[17]

P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127.  Google Scholar

[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.  Google Scholar

[19]

W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

V. I. Arnold, "Mathematial Methods of Classical Mechanics," 2nd edition, Springer, New York, 1989.  Google Scholar

[2]

A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification," Chapman $&$ Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. do Carmo, "Riemannian Geometry," Birkhaser, Boston, 1992.  Google Scholar

[9]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Milnor, "Morse Theory," Princenton University Press, Princenton, 1963.  Google Scholar

[13]

J. Moser, "Selected Chapters in the Calculus of Variations," Birkhäuser, Basel, 2003.  Google Scholar

[14]

P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. H. Poincaré, 6 (1989), 311-346.  Google Scholar

[15]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  Google Scholar

[16]

P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems, Ergodic Theory Dynam. Systems, 20 (2000), 1767-1784. doi: 10.1017/S0143385700000985.  Google Scholar

[17]

P. H. Rabinowitz, Variational methods for Hamiltonian systems, in "Handbood of Dynamical Sysstems," Vol. 1A, Elsevier, Amsterdam, (2002), 1091-1127.  Google Scholar

[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.  Google Scholar

[19]

W. Rudin, "Real and Complex Analysis," 3rd edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[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.  Google Scholar

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