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The reversibility problem for quasi-homogeneous dynamical systems
1. | Department of Mathematics, University of Huelva, 21071-Huelva, Spain, Spain |
2. | Department of Applied Mathematics II, University of Seville, 41092-Seville, Spain |
References:
[1] |
A. Algaba, E. Freire and E Gamero, Isochronicity via normal form, Qualitative Theory of Dynamical Systems, 1 (2000), 133-156.
doi: 10.1007/BF02969475. |
[2] |
A. Algaba, E. Freire, E. Gamero and C. García, Monodromy, center-focus and integrability problem for quasi-homogeneous polynomial systems, Nonlinear Analysis: Theory, Methods and Applications, 72 (2010), 1726-1736.
doi: 10.1016/j.na.2009.09.012. |
[3] |
A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
[4] |
A. Algaba, C. García and M. Reyes, Integrability of two-dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain Journal of Mathematics, 41 (2011), 1-22.
doi: 10.1216/RMJ-2011-41-1-1. |
[5] |
A. Algaba, C. García and M. A. Teixeira, Reversibility and quasi-homogeneous normal forms of vector fields, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 510-525.
doi: 10.1016/j.na.2010.03.046. |
[6] |
M. Berthier, D. Cerveau and A. Lins Neto, Sur les feuilletages analytiques réels et le problème du centre, Journal of Differential Equations, 131 (1996), 244-266.
doi: 10.1006/jdeq.1996.0163. |
[7] |
M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Annales de l'Institut Fourier, 44 (1994), 465-494. |
[8] |
A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer, Berlin, 1989.
doi: 10.1007/978-3-642-61314-2. |
[9] |
A. D. Bruno, Analysis of the Euler-Poisson equations by methods of power geometry and normal form, Journal of Applied Mathematics and Mechanics, 71 (2007), 168-199.
doi: 10.1016/j.jappmathmech.2007.06.002. |
[10] |
J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory and Dynamical Systems, 23 (2003), 417-428.
doi: 10.1017/S014338570200127X. |
[11] |
S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields," Cambridge University Press, 1994.
doi: 10.1017/CBO9780511665639. |
[12] |
J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, Berlin, 1983. |
[13] |
J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physica D. Nonlinear Phenomena, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[14] |
J. Llibre and J. C. Medrado, Darboux integrability and reversible quadratic vector fields, Rocky Mountain Journal of Mathematics, 35 (2005), 1999-2057.
doi: 10.1216/rmjm/1181069627. |
[15] |
M. V. Matveyev, Lyapunov stability of equilibrium states of reversibles systems, Mathematical Notes, 57 (1993), 63-72. |
[16] |
M. V. Matveyev, Reversible systems with first integrals, Physica D, 112 (1998), 148-157.
doi: 10.1016/S0167-2789(97)00208-X. |
[17] |
D. Montgomery and L. Zippin, "Topological Transformation Groups," Interscience Publ., New York, 1955. |
[18] |
H. Poincaré, "Mémoire sur les Courbes Définies par les Équations Différentielles," Oeuvres de Henri Poincaré, vol. 1, Gauthiers-Villars, Paris, 1951. |
[19] |
E. Strozyna and H. Zoladek, The anallytic and formal normal form for the nilpotent singularity, Journal of Differential Equations, 179 (2002), 479-537.
doi: 10.1006/jdeq.2001.4043. |
[20] |
M. A. Teixeira and J. Yang, The center-focus problem and reversibility, Journal of Differential Equations, 174 (2001), 237-251.
doi: 10.1006/jdeq.2000.3931. |
[21] |
V. N. Tkhai, The reversibility of mechanical systems, Journal of Applied Mathematics and Mechanics, 55 (1991), 461-469.
doi: 10.1016/0021-8928(91)90007-H. |
[22] |
H. Żoładek, The classification of reversible cubic systems with center, Topological Methods in Nonlinear Analysis, 4 (1994), 79-136. |
[23] |
H. Żoładek and J. Llibre, The Poincaré center problem, Journal of Dynamical and Control Systems, 14 (2008), 505-535.
doi: 10.1007/s10883-008-9049-5. |
show all references
References:
[1] |
A. Algaba, E. Freire and E Gamero, Isochronicity via normal form, Qualitative Theory of Dynamical Systems, 1 (2000), 133-156.
doi: 10.1007/BF02969475. |
[2] |
A. Algaba, E. Freire, E. Gamero and C. García, Monodromy, center-focus and integrability problem for quasi-homogeneous polynomial systems, Nonlinear Analysis: Theory, Methods and Applications, 72 (2010), 1726-1736.
doi: 10.1016/j.na.2009.09.012. |
[3] |
A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
[4] |
A. Algaba, C. García and M. Reyes, Integrability of two-dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain Journal of Mathematics, 41 (2011), 1-22.
doi: 10.1216/RMJ-2011-41-1-1. |
[5] |
A. Algaba, C. García and M. A. Teixeira, Reversibility and quasi-homogeneous normal forms of vector fields, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 510-525.
doi: 10.1016/j.na.2010.03.046. |
[6] |
M. Berthier, D. Cerveau and A. Lins Neto, Sur les feuilletages analytiques réels et le problème du centre, Journal of Differential Equations, 131 (1996), 244-266.
doi: 10.1006/jdeq.1996.0163. |
[7] |
M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Annales de l'Institut Fourier, 44 (1994), 465-494. |
[8] |
A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer, Berlin, 1989.
doi: 10.1007/978-3-642-61314-2. |
[9] |
A. D. Bruno, Analysis of the Euler-Poisson equations by methods of power geometry and normal form, Journal of Applied Mathematics and Mechanics, 71 (2007), 168-199.
doi: 10.1016/j.jappmathmech.2007.06.002. |
[10] |
J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory and Dynamical Systems, 23 (2003), 417-428.
doi: 10.1017/S014338570200127X. |
[11] |
S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields," Cambridge University Press, 1994.
doi: 10.1017/CBO9780511665639. |
[12] |
J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, Berlin, 1983. |
[13] |
J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physica D. Nonlinear Phenomena, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[14] |
J. Llibre and J. C. Medrado, Darboux integrability and reversible quadratic vector fields, Rocky Mountain Journal of Mathematics, 35 (2005), 1999-2057.
doi: 10.1216/rmjm/1181069627. |
[15] |
M. V. Matveyev, Lyapunov stability of equilibrium states of reversibles systems, Mathematical Notes, 57 (1993), 63-72. |
[16] |
M. V. Matveyev, Reversible systems with first integrals, Physica D, 112 (1998), 148-157.
doi: 10.1016/S0167-2789(97)00208-X. |
[17] |
D. Montgomery and L. Zippin, "Topological Transformation Groups," Interscience Publ., New York, 1955. |
[18] |
H. Poincaré, "Mémoire sur les Courbes Définies par les Équations Différentielles," Oeuvres de Henri Poincaré, vol. 1, Gauthiers-Villars, Paris, 1951. |
[19] |
E. Strozyna and H. Zoladek, The anallytic and formal normal form for the nilpotent singularity, Journal of Differential Equations, 179 (2002), 479-537.
doi: 10.1006/jdeq.2001.4043. |
[20] |
M. A. Teixeira and J. Yang, The center-focus problem and reversibility, Journal of Differential Equations, 174 (2001), 237-251.
doi: 10.1006/jdeq.2000.3931. |
[21] |
V. N. Tkhai, The reversibility of mechanical systems, Journal of Applied Mathematics and Mechanics, 55 (1991), 461-469.
doi: 10.1016/0021-8928(91)90007-H. |
[22] |
H. Żoładek, The classification of reversible cubic systems with center, Topological Methods in Nonlinear Analysis, 4 (1994), 79-136. |
[23] |
H. Żoładek and J. Llibre, The Poincaré center problem, Journal of Dynamical and Control Systems, 14 (2008), 505-535.
doi: 10.1007/s10883-008-9049-5. |
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