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Periodic perturbation of quadratic systems with two infinite heteroclinic cycles
1. | Departamento de Matemática, Estatística e Computação, Faculdade de Ciências e Tecnologia, Univ Estadual Paulista - UNESP, Cx.Postal 266, 19060-900, Presidente Prudente, SP, Brazil |
References:
[1] |
T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104 (1993), 215-242. |
[2] |
C. Chicone, "Ordinary Differential Equations with Applications," Texts in Appl. Math., 34, Springer-Verlag, New York, 1999. |
[3] |
C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane, J. Differential Equations, 61 (1986), 398-418.
doi: 10.1016/0022-0396(86)90113-0. |
[4] |
S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373.
doi: 10.1016/0022-0396(80)90104-7. |
[5] |
W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[6] |
H. Dankowicz and P. Holmes, The existence of transverse homoclinic points in the Sitnikov problem, J. Differential Equations, 116 (1995), 468-483.
doi: 10.1006/jdeq.1995.1044. |
[7] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[8] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. |
[9] |
A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles, J. Math. Anal. Appl., 269 (2002), 332-351.
doi: 10.1016/S0022-247X(02)00027-6. |
[10] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Appl. Math. Sci., 42, Springer-Verlag, New York, 1990. |
[11] |
J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation, Nonlinear Anal., 2 (1978), 77-84.
doi: 10.1016/0362-546X(78)90043-3. |
[12] |
I. D. Iliev, Chengzhi Li and Jiang Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.
doi: 10.1088/0951-7715/18/1/016. |
[13] |
V. K. Mel'nikov, On the stability of the center for time periodic perturbations, Trudy Moskov. Mat. Obšč., 12 (1963), 3-52. |
[14] |
M. Messias, Periodic perturbations of quadratic planar polynomial vector fields, An. Acad. Brasil. Ciênc., 74 (2002), 193-198. |
[15] |
M. Messias, Subharmonic bifurcations near infinity, Qual. Theory Dyn. Syst., 5 (2004), 301-336.
doi: 10.1007/BF02972684. |
[16] |
C. Rousseau and H. Zhu, PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem, J. Differential Equations, 196 (2004), 169-208. |
[17] |
J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane, in "Oscillations, Bifurcation and Chaos" (Toronto, Ont., 1986), CMS Conf. Proc., 8, Amer. Math. Soc., Providence, RI, (1987), 665-685. |
[18] |
P. Táboas, Periodic solutions of a forced Lotka-Volterra equation, J. Math. Anal. Appl., 124 (1987), 82-97.
doi: 10.1016/0022-247X(87)90026-6. |
[19] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Texts in Appl. Math., 2, Springer-Verlag, New York, 1990. |
show all references
References:
[1] |
T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104 (1993), 215-242. |
[2] |
C. Chicone, "Ordinary Differential Equations with Applications," Texts in Appl. Math., 34, Springer-Verlag, New York, 1999. |
[3] |
C. Chicone and J. Sotomayor, On a class of complete polynomial vector fields in the plane, J. Differential Equations, 61 (1986), 398-418.
doi: 10.1016/0022-0396(86)90113-0. |
[4] |
S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373.
doi: 10.1016/0022-0396(80)90104-7. |
[5] |
W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[6] |
H. Dankowicz and P. Holmes, The existence of transverse homoclinic points in the Sitnikov problem, J. Differential Equations, 116 (1995), 468-483.
doi: 10.1006/jdeq.1995.1044. |
[7] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[8] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Theory of Planar Differential Systems," Universitext, Springer-Verlag, Berlin, 2006. |
[9] |
A. Gasull, V. Mañosa and F. Mañosas, Stability of certain planar unbounded polycycles, J. Math. Anal. Appl., 269 (2002), 332-351.
doi: 10.1016/S0022-247X(02)00027-6. |
[10] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Appl. Math. Sci., 42, Springer-Verlag, New York, 1990. |
[11] |
J. Hale and P. Táboas, Interaction of damping and forcing in a second order equation, Nonlinear Anal., 2 (1978), 77-84.
doi: 10.1016/0362-546X(78)90043-3. |
[12] |
I. D. Iliev, Chengzhi Li and Jiang Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.
doi: 10.1088/0951-7715/18/1/016. |
[13] |
V. K. Mel'nikov, On the stability of the center for time periodic perturbations, Trudy Moskov. Mat. Obšč., 12 (1963), 3-52. |
[14] |
M. Messias, Periodic perturbations of quadratic planar polynomial vector fields, An. Acad. Brasil. Ciênc., 74 (2002), 193-198. |
[15] |
M. Messias, Subharmonic bifurcations near infinity, Qual. Theory Dyn. Syst., 5 (2004), 301-336.
doi: 10.1007/BF02972684. |
[16] |
C. Rousseau and H. Zhu, PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem, J. Differential Equations, 196 (2004), 169-208. |
[17] |
J. Sotomayor and R. Paterlini, Bifurcation of polynomial vector fields in the plane, in "Oscillations, Bifurcation and Chaos" (Toronto, Ont., 1986), CMS Conf. Proc., 8, Amer. Math. Soc., Providence, RI, (1987), 665-685. |
[18] |
P. Táboas, Periodic solutions of a forced Lotka-Volterra equation, J. Math. Anal. Appl., 124 (1987), 82-97.
doi: 10.1016/0022-247X(87)90026-6. |
[19] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Texts in Appl. Math., 2, Springer-Verlag, New York, 1990. |
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