Formal equivalence between normal forms of reversible and hamiltonian dynamical systems

Ricardo Miranda Martins

doi:10.3934/cpaa.2014.13.703

Article Contents

2014, Volume 13, Issue 2: 703-713. Doi: 10.3934/cpaa.2014.13.703

This issue Previous Article Topological conjugacies and behavior at infinity Next Article Lifespan theorem and gap lemma for the globally constrained Willmore flow

Formal equivalence between normal forms of reversible and hamiltonian dynamical systems

Ricardo Miranda Martins^1,

1.
Instituto de Matemática, Estatística e Computação Cientíﬁca, Universidade Estadual de Campinas, 13083–859 Campinas, SP

Received: February 28, 2013

Revised: July 31, 2013

Published: February 2014

Abstract Related Papers Cited by

Abstract

We show the existence of formal equivalences between $2n$-dimensional reversible and Hamiltonian vector fields. The main tool we employ is the normal form theory.

Keywords:

Mathematics Subject Classification: Primary: 34C14, 34C20; Secondary: 37J15, 37J40.

Citation:

Related Papers

Cited by

References

[1]	R. Abraham and J. Marsden, "Foundations of Mechanics," Benjamin Cummings, London, 1978.
[2]	F. Antoneli, P. H. Baptistelli, A. P. Dias and M. Manoel, Invariant theory and reversible-equivariant vector fields, J. Pure Appl. Algebra, 213 (2009), 649-663.doi: 10.1016/j.jpaa.2008.08.002.
[3]	V. I. Arnold, "Arnold's Problems," Springer-Verlag, Berlin, 2004.
[4]	V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Fundamental Principles of Mathematical Sciences 250. Springer-Verlag, 1998.
[5]	G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300.doi: 10.1090/S0002-9947-1917-1501070-3.
[6]	R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.doi: 10.1090/S0002-9947-1976-0402815-3.
[7]	G. Gaeta, Normal forms of reversible dynamical systems, International Journal of Theoretical Physics, 33 (1994), 1917-1928.doi: 10.1007/BF00671033.
[8]	A. Jacquemard, M. F. S. Lima and M. A. Teixeira, Degenerate resonances and branching of periodic orbits, Annali di Matematica Pura ed Applicata, 187 (2008), 105-117.doi: 10.1007/s10231-006-0036-8.
[9]	R. M. Martins and M. A. Teixeira, On the Similarity of Hamiltonian and reversible vector fields in 4D, Communications on Pure and Applied Analysis, 10 (2011), 1257-1266.doi: 10.3934/cpaa.2011.10.1257.
[10]	R. M. Martins and M. A. Teixeira, Reversible-equivariant systems and matricial equations, Anais da Academia Brasileira de Ciências, 83 (2011), 1-16.doi: 10.1590/S0001-37652011000200003.
[11]	J. C. van der Meer, "The Hamiltonian Hopf Bifurcation," Lecture Notes in Mathematics, 1160, Springer Berlin, 1982.
[12]	J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, "Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance," Dynamics, Bifurcation and Symmetry: New Trends and New Tools, Kluwer Academic Publishers, 1994.
[13]	G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79.doi: 10.1090/S0002-9947-1935-1501778-0.
[14]	M. B. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1935), 132-147.doi: 10.1016/S0167-2789(97)00207-8.
[15]	M. A. Teixeira, Singularities of reversible vector fields, Phys. D, 100 (1997), 101-118.doi: 10.1016/S0167-2789(96)00183-2.