Formal equivalence between normal forms of reversible and hamiltonian dynamical systems

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March 2014, 13(2): 703-713. doi: 10.3934/cpaa.2014.13.703

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 March 2013 Revised August 2013 Published October 2013

Abstract
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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: normal forms., Hamiltonian systems, Reversible vector fields.

Mathematics Subject Classification: Primary: 34C14, 34C20; Secondary: 37J15, 37J4.

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

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.

show all references

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.

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