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

The reversibility problem for quasi-homogeneous dynamical systems

Antonio Algaba 1, , Estanislao Gamero 2, and  Cristóbal García 1,

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.
Keywords: Lie symmetries., Reversibility problem, quasi-homogeneous vector fields, Center problem.
Mathematics Subject Classification: Primary: 34C14; Secondary: 34A2.
Citation: Antonio Algaba, Estanislao Gamero, Cristóbal García. The reversibility problem for quasi-homogeneous dynamical systems. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Annales de l'Institut Fourier, 44 (1994), 465-494.  Google Scholar

[8]

A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer, Berlin, 1989. doi: 10.1007/978-3-642-61314-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. V. Matveyev, Lyapunov stability of equilibrium states of reversibles systems, Mathematical Notes, 57 (1993), 63-72. Google Scholar

[16]

M. V. Matveyev, Reversible systems with first integrals, Physica D, 112 (1998), 148-157. doi: 10.1016/S0167-2789(97)00208-X.  Google Scholar

[17]

D. Montgomery and L. Zippin, "Topological Transformation Groups," Interscience Publ., New York, 1955.  Google Scholar

[18]

H. Poincaré, "Mémoire sur les Courbes Définies par les Équations Différentielles," Oeuvres de Henri Poincaré, vol. 1, Gauthiers-Villars, Paris, 1951. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. Żoładek, The classification of reversible cubic systems with center, Topological Methods in Nonlinear Analysis, 4 (1994), 79-136.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Annales de l'Institut Fourier, 44 (1994), 465-494.  Google Scholar

[8]

A. D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer, Berlin, 1989. doi: 10.1007/978-3-642-61314-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, Berlin, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. V. Matveyev, Lyapunov stability of equilibrium states of reversibles systems, Mathematical Notes, 57 (1993), 63-72. Google Scholar

[16]

M. V. Matveyev, Reversible systems with first integrals, Physica D, 112 (1998), 148-157. doi: 10.1016/S0167-2789(97)00208-X.  Google Scholar

[17]

D. Montgomery and L. Zippin, "Topological Transformation Groups," Interscience Publ., New York, 1955.  Google Scholar

[18]

H. Poincaré, "Mémoire sur les Courbes Définies par les Équations Différentielles," Oeuvres de Henri Poincaré, vol. 1, Gauthiers-Villars, Paris, 1951. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. Żoładek, The classification of reversible cubic systems with center, Topological Methods in Nonlinear Analysis, 4 (1994), 79-136.  Google Scholar

[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.  Google Scholar

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