November  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, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[1]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[2]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[3]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[4]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[5]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089

[6]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023

[7]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024

[8]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3579. doi: 10.3934/dcdsb.2020246

[9]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099

[10]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[11]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[12]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405

[13]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[14]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

[15]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198

[16]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[17]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071

[18]

Zheng Liu, Tianxiao Wang. A class of stochastic Fredholm-algebraic equations and applications in finance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3879-3903. doi: 10.3934/dcdsb.2020267

[19]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018

[20]

Roberto Civino, Riccardo Longo. Formal security proof for a scheme on a topological network. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021009

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]