American Institute of Mathematical Sciences

Advanced
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

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, London, 1978.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Fundamental Principles of Mathematical Sciences 250. Springer-Verlag, 1998.  Google Scholar

[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.  Google Scholar

[6]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. C. van der Meer, "The Hamiltonian Hopf Bifurcation," Lecture Notes in Mathematics, 1160, Springer Berlin, 1982.  Google Scholar

[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.  Google Scholar

[13]

G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.1090/S0002-9947-1935-1501778-0.  Google Scholar

[14]

M. B. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1935), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[15]

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]

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.  Google Scholar

[3]

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

[4]

V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Fundamental Principles of Mathematical Sciences 250. Springer-Verlag, 1998.  Google Scholar

[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.  Google Scholar

[6]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. C. van der Meer, "The Hamiltonian Hopf Bifurcation," Lecture Notes in Mathematics, 1160, Springer Berlin, 1982.  Google Scholar

[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.  Google Scholar

[13]

G. B. Price, On reversible dynamical systems, Trans. Amer. Math. Soc., 37 (1935), 51-79. doi: 10.1090/S0002-9947-1935-1501778-0.  Google Scholar

[14]

M. B. Sevryuk, The finite-dimensional reversible KAM theory, Phys. D, 112 (1935), 132-147. doi: 10.1016/S0167-2789(97)00207-8.  Google Scholar

[15]

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

[1]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
[2]

Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769
[3]

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
[4]

Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092
[5]

Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
[6]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
[7]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
[8]

Charles-Michel Marle. A property of conformally Hamiltonian vector fields; Application to the Kepler problem. Journal of Geometric Mechanics, 2012, 4 (2) : 181-206. doi: 10.3934/jgm.2012.4.181
[9]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[10]

André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359
[11]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225
[12]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092
[13]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65
[14]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[15]

Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227
[16]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[17]

A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.

[18]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1
[19]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014
[20]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy