## Abstract and Applied Analysis

### The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

#### Abstract

We consider the existence of the periodic solutions in the neighbourhood of equilibria for ${C}^{\infty }$ equivariant Hamiltonian vector fields. If the equivariant symmetry $S$ acts antisymplectically and ${S}^{2}=I$, we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds each containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian systems.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 530209, 12 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495630

Digital Object Identifier
doi:10.1155/2012/530209

Mathematical Reviews number (MathSciNet)
MR2872320

Zentralblatt MATH identifier
1239.34049

#### Citation

Li, Jia; Shi, Yanling. The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems. Abstr. Appl. Anal. 2012 (2012), Article ID 530209, 12 pages. doi:10.1155/2012/530209. https://projecteuclid.org/euclid.aaa/1355495630

#### References

• W. B. Gordon, “A theorem on the existence of periodic solutions to Hamiltonian systems with convex potential,” Journal of Differential Equations, vol. 10, pp. 324–335, 1971.
• A. Weinstein, “Normal modes for nonlinear Hamiltonian systems,” Inventiones Mathematicae, vol. 20, pp. 47–57, 1973.
• J. Moser, “Periodic orbits near an equilibrium and a theorem by Alan Weinstein,” Communications on Pure and Applied Mathematics, vol. 29, no. 6, pp. 724–747, 1976.
• A. Weinstein, “Periodic orbits for convex Hamiltonian systems,” Annals of Mathematics, vol. 108, no. 3, pp. 507–518, 1978.
• J.-C. van der Meer, The Hamiltonian Hopf Bifurcation, vol. 1160 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1985.
• M. Golubitsky and W. F. Langford, “Classification and unfoldings of degenerate Hopf bifurcations,” Journal of Differential Equations, vol. 41, no. 3, pp. 375–415, 1981.
• T. J. Bridges, “Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, no. 3, pp. 575–601, 1990.
• R. L. Devaney, “Reversible diffeomorphisms and flows,” Transactions of the American Mathematical Society, vol. 218, pp. 89–113, 1976.
• A. Vanderbauwhede, Local Bifurcation and Symmetry, vol. 75 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1982.
• M. B. Sevryuk, Reversible Systems, vol. 1211 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.
• M. Golubitsky, M. Krupa, and C. Lim, “Time-reversibility and particle sedimentation,” SIAM Journal on Applied Mathematics, vol. 51, no. 1, pp. 49–72, 1991.
• J. A. Montaldi, R. M. Roberts, and I. N. Stewart, “Periodic solutions near equilibria of symmetric Hamiltonian systems,” Philosophical Transactions of the Royal Society of London A, vol. 325, no. 1584, pp. 237–293, 1988.
• J. Montaldi, M. Roberts, and I. Stewart, “Existence of nonlinear normal modes of symmetric Hamiltonian systems,” Nonlinearity, vol. 3, no. 3, pp. 695–730, 1990.
• J. S. W. Lamb and M. Roberts, “Reversible equivariant linear systems,” Journal of Differential Equations, vol. 159, no. 1, pp. 239–279, 1999.
• I. Hoveijn, J. S. W. Lamb, and R. M. Roberts, “Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two,” Journal of Differential Equations, vol. 190, no. 1, pp. 182–213, 2003.
• C. A. Buzzi and J. S. W. Lamb, “Reversible Hamiltonian Liapunov center theorem,” Discrete and Continuous Dynamical Systems B, vol. 5, no. 1, pp. 51–66, 2005.
• G. R. Belitskii and A. Y. Kopanskii, “Sternberg theorem for equivariant Hamiltonian vector fields,” Nonlinear Analysis, vol. 47, no. 7, pp. 4491–4499, 2001.
• M. Golubitsky, J. E. Marsden, I. Stewart, and M. Dellnitz, “The constrained Liapunov-Schmidt procedure and periodic orbits,” Fields Institute Communications, vol. 4, pp. 81–127, 1995.
• M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. Vol. I, vol. 51 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1985.
• R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1988.
• R. Abraham and J. E. Marsden, Foundations of Mechanics, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1978.