# American Institute of Mathematical Sciences

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

## Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

 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.
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.
 [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 & 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 & 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 & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021 [5] Andrea Venturelli. A Variational proof of the existence of Von Schubart's orbit. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 699-717. doi: 10.3934/dcdsb.2008.10.699 [6] Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 [7] Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098 [8] Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799 [9] 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 [10] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [11] Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769 [12] Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883 [13] Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507 [14] Zheng Yin, Ercai Chen. The conditional variational principle for maps with the pseudo-orbit tracing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 463-481. doi: 10.3934/dcds.2019019 [15] 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 [16] Ling-Hao Zhang, Wei Wang. Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 591-604. doi: 10.3934/dcds.2017024 [17] Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039 [18] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 [19] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [20] Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

2017 Impact Factor: 1.179