February  2005, 5(1): 51-66. doi: 10.3934/dcdsb.2005.5.51

Reversible Hamiltonian Liapunov center theorem

1. 

IBILCE, UNESP, São José do Rio Preto, CEP 15054-000, Brazil

2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  September 2003 Revised  February 2004 Published  November 2004

We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry $R$. We contrast the cases where $R$ acts symplectically and anti-symplectically.
In case $R$ acts anti-symplectically, generically purely imaginary eigenvalues are isolated, and the equilibrium is contained in a local two-dimensional invariant manifold containing symmetric periodic solutions encircling the equilibrium point.
In case $R$ acts symplectically, generically purely imaginary eigenvalues are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant manifolds containing nonsymmetric periodic solutions encircling the equilibrium point. In addition, there exists a three-dimensional invariant surface containing a two-parameter family of symmetric periodic solutions.
Citation: Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[1]

Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems and Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63

[2]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

[3]

Rodrigo I. Brevis, Jaime H. Ortega, David Pardo. A source time reversal method for seismicity induced by mining. Inverse Problems and Imaging, 2017, 11 (1) : 25-45. doi: 10.3934/ipi.2017002

[4]

Albert Fannjiang, Knut Solna. Time reversal of parabolic waves and two-frequency Wigner distribution. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 783-802. doi: 10.3934/dcdsb.2006.6.783

[5]

Kazufumi Ito, Karim Ramdani, Marius Tucsnak. A time reversal based algorithm for solving initial data inverse problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 641-652. doi: 10.3934/dcdss.2011.4.641

[6]

Saroj Panigrahi. Liapunov-type integral inequalities for higher order dynamic equations on time scales. Conference Publications, 2013, 2013 (special) : 629-641. doi: 10.3934/proc.2013.2013.629

[7]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213

[8]

Corinna Burkard, Aurelia Minut, Karim Ramdani. Far field model for time reversal and application to selective focusing on small dielectric inhomogeneities. Inverse Problems and Imaging, 2013, 7 (2) : 445-470. doi: 10.3934/ipi.2013.7.445

[9]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[10]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[11]

Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006

[12]

Lijun Zhang, Xiangshuo Liu, Chaohong Pan. Studies on reversal permanent charges and reversal potentials via classical Poisson-Nernst-Planck systems with boundary layers. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022013

[13]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[14]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593

[15]

Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296

[16]

J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

[17]

Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004

[18]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[19]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89

[20]

B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]