American Institute of Mathematical Sciences

Advanced
January  2013, 33(1): 359-363. doi: 10.3934/dcds.2013.33.359

Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems

André Vanderbauwhede 1,

1. 

Department of Pure Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium

Received  August 2011 Revised  February 2012 Published  September 2012

In this paper we discuss some general results on families of symmetric and doubly-symmetric solutions in reversible Hamiltonian systems having several independent first integrals. We describe a set-up for such solutions which allows the application of classical continuation and bifurcation results.
Keywords: Continuation, reversible Hamiltonian systems, multi-symmetric solutions., bifurcation.
Mathematics Subject Classification: Primary: 37J45, 37G40; Secondary: 37J4.
Citation: André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359
References:
[1]

F. J. Muñoz-Almaraz, E. Freire, J. Galán, E. J. Doedel and A. Vanderbauwhede, Continuation of periodic orbits in conservative and Hamiltonian systems,, Physica D, 181 (2003), 1.  doi: 10.1016/S0167-2789(03)00097-6.  Google Scholar

[2]

F. J. Muñoz-Almaraz , E. Freire, J. Galán, and A. Vanderbauwhede, Continuation of Gerver's supereight choreography,, Monografias de la Real Academia de Ciencias de Zaragoza, 30 (2006), 95.   Google Scholar

[3]

F. J. Muñoz-Almaraz, E. Freire, J. Galán and A. Vanderbauwhede, Continuation of normal doubly symmetric orbits in conservative reversible systems,, Celestial Mechanics & Dynamical Astronomy, 97 (2007), 17.   Google Scholar

[4]

, http://www.maia.ub.es/ malmaraz/investigacion/Jaca/jaca.xml., ().   Google Scholar

show all references

References:
[1]

F. J. Muñoz-Almaraz, E. Freire, J. Galán, E. J. Doedel and A. Vanderbauwhede, Continuation of periodic orbits in conservative and Hamiltonian systems,, Physica D, 181 (2003), 1.  doi: 10.1016/S0167-2789(03)00097-6.  Google Scholar

[2]

F. J. Muñoz-Almaraz , E. Freire, J. Galán, and A. Vanderbauwhede, Continuation of Gerver's supereight choreography,, Monografias de la Real Academia de Ciencias de Zaragoza, 30 (2006), 95.   Google Scholar

[3]

F. J. Muñoz-Almaraz, E. Freire, J. Galán and A. Vanderbauwhede, Continuation of normal doubly symmetric orbits in conservative reversible systems,, Celestial Mechanics & Dynamical Astronomy, 97 (2007), 17.   Google Scholar

[4]

, http://www.maia.ub.es/ malmaraz/investigacion/Jaca/jaca.xml., ().   Google Scholar

[1]

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
[2]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703
[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]

Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337
[5]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[6]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757
[7]

Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227
[8]

Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations. Networks & Heterogeneous Media, 2019, 14 (3) : 567-587. doi: 10.3934/nhm.2019022
[9]

Péter Bálint, Imre Péter Tóth. Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 37-59. doi: 10.3934/dcds.2006.15.37
[10]

Paul H. Rabinowitz. On a class of reversible elliptic systems. Networks & Heterogeneous Media, 2012, 7 (4) : 927-939. doi: 10.3934/nhm.2012.7.927
[11]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[12]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069
[13]

Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467
[14]

Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059
[15]

Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937
[16]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
[17]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047
[18]

Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
[19]

José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357
[20]

Yuming Qin, Jianlin Zhang. Global existence and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2529-2574. doi: 10.3934/cpaa.2019115

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences