2003, 9(6): 1493-1518. doi: 10.3934/dcds.2003.9.1493

Global bifurcation of homoclinic solutions of Hamiltonian systems

1. 

Scuola Internazionale Superiore de Studi Avanzati, via Beirut 2-4, 34014 Trieste, Italy

2. 

Institut de Analyse et de Calcul Scientifique, Section de mathmatiques, Ecole Polytechnique Fédérale Lausanne, CH - 1015 Lausanne, Switzerland

Received  July 2002 Revised  April 2003 Published  September 2003

The main results give hypotheses ensuring that a non-autonomous first order Hamiltonian system has a global branch of homoclinic solutions bifurcating from an eigenvalue of odd multiplicity of the linearization. The system is required to be asymptotically periodic (as time goes to plus and minus infinity) and these limit problems should have no homoclinic solutions. Furthermore, the asymptotic limits of the linearization should have no characteristic multipliers on the unit circle. The proof uses the topological degree for proper Fredholm maps of index zero.
Citation: S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[1]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

[2]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

[3]

Lora Billings, Erik M. Bollt, David Morgan, Ira B. Schwartz. Stochastic global bifurcation in perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 123-132. doi: 10.3934/proc.2003.2003.123

[4]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[5]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471

[6]

Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013

[7]

James Benn. Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group. Journal of Geometric Mechanics, 2016, 8 (1) : 1-12. doi: 10.3934/jgm.2016.8.1

[8]

Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang. On the global optimal solution for linear quadratic problems of switched system. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-16. doi: 10.3934/jimo.2018072

[9]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885

[10]

Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237

[11]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

[12]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971

[13]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419

[14]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295

[15]

Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669

[16]

Hideaki Takaichi, Izumi Takagi, Shoji Yotsutani. Global bifurcation structure on a shadow system with a source term - Representation of all solutions-. Conference Publications, 2011, 2011 (Special) : 1344-1350. doi: 10.3934/proc.2011.2011.1344

[17]

Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347

[18]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[19]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[20]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]