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

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 & 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, (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. doi: 10.1016/j.jpaa.2008.08.002.
[3]	V. I. Arnold, "Arnold's Problems,", Springer-Verlag, (2004).
[4]	V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Fundamental Principles of Mathematical Sciences {\bf 250}. Springer-Verlag, 250 (1998).
[5]	G. D. Birkhoff, Dynamical systems with two degrees of freedom,, Trans. Amer. Math. Soc., 18 (1917), 199. doi: 10.1090/S0002-9947-1917-1501070-3.
[6]	R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Amer. Math. Soc., 218 (1976), 89. 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. 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. 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. 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 Ci\^encias, 83 (2011), 1. doi: 10.1590/S0001-37652011000200003.
[11]	J. C. van der Meer, "The Hamiltonian Hopf Bifurcation,", Lecture Notes in Mathematics, 1160 (1982).
[12]	J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, "Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance,", Dynamics, (1994).
[13]	G. B. Price, On reversible dynamical systems,, Trans. Amer. Math. Soc., 37 (1935), 51. doi: 10.1090/S0002-9947-1935-1501778-0.
[14]	M. B. Sevryuk, The finite-dimensional reversible KAM theory,, Phys. D, 112 (1935), 132. doi: 10.1016/S0167-2789(97)00207-8.
[15]	M. A. Teixeira, Singularities of reversible vector fields,, Phys. D, 100 (1997), 101. doi: 10.1016/S0167-2789(96)00183-2.

show all references

References:

[1]	R. Abraham and J. Marsden, "Foundations of Mechanics,", Benjamin Cummings, (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. doi: 10.1016/j.jpaa.2008.08.002.
[3]	V. I. Arnold, "Arnold's Problems,", Springer-Verlag, (2004).
[4]	V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Fundamental Principles of Mathematical Sciences {\bf 250}. Springer-Verlag, 250 (1998).
[5]	G. D. Birkhoff, Dynamical systems with two degrees of freedom,, Trans. Amer. Math. Soc., 18 (1917), 199. doi: 10.1090/S0002-9947-1917-1501070-3.
[6]	R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Amer. Math. Soc., 218 (1976), 89. 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. 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. 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. 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 Ci\^encias, 83 (2011), 1. doi: 10.1590/S0001-37652011000200003.
[11]	J. C. van der Meer, "The Hamiltonian Hopf Bifurcation,", Lecture Notes in Mathematics, 1160 (1982).
[12]	J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, "Hamiltonian Structure of the Reversible Nonsemisimple 1:1 Resonance,", Dynamics, (1994).
[13]	G. B. Price, On reversible dynamical systems,, Trans. Amer. Math. Soc., 37 (1935), 51. doi: 10.1090/S0002-9947-1935-1501778-0.
[14]	M. B. Sevryuk, The finite-dimensional reversible KAM theory,, Phys. D, 112 (1935), 132. doi: 10.1016/S0167-2789(97)00207-8.
[15]	M. A. Teixeira, Singularities of reversible vector fields,, Phys. D, 100 (1997), 101. doi: 10.1016/S0167-2789(96)00183-2.

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