Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

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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

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

References:

[1]	V. I. Arnold, "Mathematial Methods of Classical Mechanics,", 2nd edition, (1989).
[2]	A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification,", Chapman $&$ Hall/CRC, (2004).
[3]	K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems,", Studies in Advanced Mathematics, (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.
[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. 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. 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. doi: 10.1007/s11425-007-0156-7.
[8]	M. do Carmo, "Riemannian Geometry,", Birkhaser, (1992).
[9]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).
[10]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Diff. Eqns., 219 (2005), 375. 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. doi: 10.1016/j.jde.2007.03.013.
[12]	J. Milnor, "Morse Theory,", Princenton University Press, (1963).
[13]	J. Moser, "Selected Chapters in the Calculus of Variations,", Birkhäuser, (2003).
[14]	P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system,, Ann. Inst. H. Poincaré, 6 (1989), 311.
[15]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33.
[16]	P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems,, Ergodic Theory Dynam. Systems, 20 (2000), 1767. doi: 10.1017/S0143385700000985.
[17]	P. H. Rabinowitz, Variational methods for Hamiltonian systems,, in, 1A (2002), 1091.
[18]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 472. doi: 10.1007/BF02571356.
[19]	W. Rudin, "Real and Complex Analysis,", 3rd edition, (1987).
[20]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25. doi: 10.1006/jfan.2001.3798.

show all references

References:

[1]	V. I. Arnold, "Mathematial Methods of Classical Mechanics,", 2nd edition, (1989).
[2]	A. V. Bolsinov and A. T. Fomenko, "Integrable Hamiltonian Systems: Geometry, Topology and Classification,", Chapman $&$ Hall/CRC, (2004).
[3]	K. Burns and M. Gidea, "Differential Geometry and Topology: With a View to Dynamical Systems,", Studies in Advanced Mathematics, (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.
[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. 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. 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. doi: 10.1007/s11425-007-0156-7.
[8]	M. do Carmo, "Riemannian Geometry,", Birkhaser, (1992).
[9]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).
[10]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Diff. Eqns., 219 (2005), 375. 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. doi: 10.1016/j.jde.2007.03.013.
[12]	J. Milnor, "Morse Theory,", Princenton University Press, (1963).
[13]	J. Moser, "Selected Chapters in the Calculus of Variations,", Birkhäuser, (2003).
[14]	P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system,, Ann. Inst. H. Poincaré, 6 (1989), 311.
[15]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33.
[16]	P. H. Rabinowitz, Connecting orbits for a reversible Hamiltonian systems,, Ergodic Theory Dynam. Systems, 20 (2000), 1767. doi: 10.1017/S0143385700000985.
[17]	P. H. Rabinowitz, Variational methods for Hamiltonian systems,, in, 1A (2002), 1091.
[18]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 472. doi: 10.1007/BF02571356.
[19]	W. Rudin, "Real and Complex Analysis,", 3rd edition, (1987).
[20]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25. doi: 10.1006/jfan.2001.3798.

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