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July  2011, 10(4): 1257-1266. doi: 10.3934/cpaa.2011.10.1257

On the similarity of Hamiltonian and reversible vector fields in 4D

Ricardo Miranda Martins 1, and  Marco Antonio Teixeira 2,

1. 

Department of Mathematics, IMECC, Unicamp, 13083-970, Campinas SP, Brazil
2. 

Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P., Brazil

Received  July 2010 Revised  September 2010 Published  April 2011

We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal form theory.
Keywords: reversible vector field, Hamiltonian vector field., Normal form.
Mathematics Subject Classification: Primary: 34C14, 34C20; Secondary: 37J15, 37J4.
Citation: Ricardo Miranda Martins, Marco Antonio Teixeira. On the similarity of Hamiltonian and reversible vector fields in 4D. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1257-1266. doi: 10.3934/cpaa.2011.10.1257
References:
[1]

R. Abraham and J. Marsden, "Foundations of Mechanics," Benjamin Cummings, London, 1978.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems," Springer-Verlag, Berlin, 2004.  Google Scholar

[3]

V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,'' Springer-Verlag, New York, 1988. Google Scholar

[4]

G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.2307/1988861.  Google Scholar

[5]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.2307/1997429.  Google Scholar

[6]

G. Gaeta, Normal Forms of reversible dynamical systems, International Journal of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[7]

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. Google Scholar

[8]

J. S. W. Lamb, M. F. S. Lima, R. M. Martins, M. A. Teixeira and J. Yang, On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in $R^4$, IMECC/Unicamp Research Report 05/10, 2010. Available from: http://www1.ime.unicamp.br/rel_pesq/relatorio.html. Google Scholar

[9]

J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, Hamiltonian Structure of the Reversible Nonsemisimple $1:1$ Resonance, in "Dynamics, Bifurcation and Symmetry: New Trends and New Tools", Kluwer Academic Publishers, (1994), 221-240.  Google Scholar

[10]

G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.2307/1989695.  Google Scholar

[11]

M. A. Teixeira, Singularities of reversible vector fields, Phys. D, 100 (1997), 101-118. doi: 10.1016/S0167-2789(96)00183-2.  Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, "Foundations of Mechanics," Benjamin Cummings, London, 1978.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems," Springer-Verlag, Berlin, 2004.  Google Scholar

[3]

V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,'' Springer-Verlag, New York, 1988. Google Scholar

[4]

G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.2307/1988861.  Google Scholar

[5]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.2307/1997429.  Google Scholar

[6]

G. Gaeta, Normal Forms of reversible dynamical systems, International Journal of Theoretical Physics, 33 (1994), 1917-1928. doi: 10.1007/BF00671033.  Google Scholar

[7]

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. Google Scholar

[8]

J. S. W. Lamb, M. F. S. Lima, R. M. Martins, M. A. Teixeira and J. Yang, On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in $R^4$, IMECC/Unicamp Research Report 05/10, 2010. Available from: http://www1.ime.unicamp.br/rel_pesq/relatorio.html. Google Scholar

[9]

J. C. van der Meer, J. A. Sanders and A. Vanderbauwhede, Hamiltonian Structure of the Reversible Nonsemisimple $1:1$ Resonance, in "Dynamics, Bifurcation and Symmetry: New Trends and New Tools", Kluwer Academic Publishers, (1994), 221-240.  Google Scholar

[10]

G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.2307/1989695.  Google Scholar

[11]

M. A. Teixeira, Singularities of reversible vector fields, Phys. D, 100 (1997), 101-118. doi: 10.1016/S0167-2789(96)00183-2.  Google Scholar

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