# American Institute of Mathematical Sciences

August  2013, 33(8): 3225-3236. doi: 10.3934/dcds.2013.33.3225

## 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

Received  May 2012 Revised  November 2012 Published  January 2013

In this paper, we obtain necessary and sufficient conditions for the reversibility of a quasi-homogeneous $n$-dimensional system. As a consequence, we get necessary conditions for an arbitrary system to be orbital-reversible. Moreover, we give sufficient conditions for orbital-reversibility in terms of the existence of Lie symmetries of the vector field. The results obtained are conveniently adapted to the case of planar systems, where we give sufficient conditions for a degenerate planar vector field to have a center at the origin. We apply the results to some case studies. Namely, we consider a family of planar vector fields, where we determine centers which are not orbital-reversible. We also study some tridimensional systems, where nonlinear involutions are determined in the reversible situations.
Citation: Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3225-3236. doi: 10.3934/dcds.2013.33.3225
##### 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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##### 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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