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Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom
Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$
1. | Department of Mathematics, Fudan University, Shanghai, 200433, China |
2. | Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275 |
References:
[1] |
M. Bobienski and H. Zoladek, Limit cycles of three-dimensional polynomial vector fields, Nonlinearity, 18 (2005), 175-209.
doi: 10.1088/0951-7715/18/1/010. |
[2] |
M. I. T. Camacho, Geometric properties of homogeneous vector fields of degree two in $R^3$, Transactions of the American Mathematical Society, 268 (1981), 79-101.
doi: 10.2307/1998338. |
[3] |
R. E. Gomory, Trajectories tending to a critical point in 3-space, Annals of Mathematics, 61 (1955), 140-153. |
[4] |
J. Huang and Y. Zhao, The limit set of trajectory in quasi-homogeneous system on $\mathbb{R}^3$, Applicable Analysis.
doi: 10.1080/00036811.2011.567193. |
[5] |
J. Huang and Y. Zhao, The projective vector field of a kind of three-dimensional quasi-homogeneous system on $\mathbbS^2$, Nonlin. Anal., 74 (2011), 4088-4104.
doi: 10.1016/j.na.2011.03.043. |
[6] |
J. Huang and Y. Zhao, Extended quasi-homogeneous polynomial system in $\mathbb{R}^3$, submitted. |
[7] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. |
[8] |
R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. |
[9] |
J. Llibre, C. A. Buzzi and P. R. da Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst., 17 (2007), 529-540. |
[10] |
J. Llibre, J. S. Perez Del Rio and J. A. Rodriguez, Structural stability of planar homogeneous polynomial vector fields: applications to critical points and to infinity, Journal of Differential Equations, 125 (1996), 490-520.
doi: 10.1006/jdeq.1996.0038. |
[11] |
J. Llibre and C. Pessoa, Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere, Rend. Circ. Mat. Palermo, 55 (2006), 63-81.
doi: 10.1007/BF02874668. |
[12] |
J. Llibre and C. Pessoa, homogeneous polynomial vector fields of degree $2$ on the $2$-dimensional sphere, Extracta Math., 21 (2006), 167-190. |
[13] |
J. Llibre and H. Wu, Hopf bifurcation for degenerate singular points of multiplicity $2n-1$ in dimension $3$, Bull. Sci. Math., 132 (2008), 218-231.
doi: 10.1016/j.bulsci.2007.01.003. |
[14] |
J. Llibre and J. Yu, Limit cycles for a class of three-dimensional polynomial differential systems, Journal of Dynamical and Control Systems, 13 (2007), 531-539.
doi: 10.1007/s10883-007-9025-5. |
[15] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[16] |
J. Llibre and X. Zhang, Hopf bifurcation in higher dimensional differential systems via the averaging method, Pacific J. Math., 240 (2009), 321-341.
doi: 10.2140/pjm.2009.240.321. |
[17] |
L. Markus, Quadratic differential equations and non-associative algebras, in "Contributions to the Theory of Non-linear Oscillations" (ed. S. Lfschetz), Princeton, Univ. Press, (1960), 185-213. |
[18] |
Y. Ye et al., "Theory of Limit Cycles," Transl. Math. Monogr. American Mathematical Society, Providence, 1986. |
show all references
References:
[1] |
M. Bobienski and H. Zoladek, Limit cycles of three-dimensional polynomial vector fields, Nonlinearity, 18 (2005), 175-209.
doi: 10.1088/0951-7715/18/1/010. |
[2] |
M. I. T. Camacho, Geometric properties of homogeneous vector fields of degree two in $R^3$, Transactions of the American Mathematical Society, 268 (1981), 79-101.
doi: 10.2307/1998338. |
[3] |
R. E. Gomory, Trajectories tending to a critical point in 3-space, Annals of Mathematics, 61 (1955), 140-153. |
[4] |
J. Huang and Y. Zhao, The limit set of trajectory in quasi-homogeneous system on $\mathbb{R}^3$, Applicable Analysis.
doi: 10.1080/00036811.2011.567193. |
[5] |
J. Huang and Y. Zhao, The projective vector field of a kind of three-dimensional quasi-homogeneous system on $\mathbbS^2$, Nonlin. Anal., 74 (2011), 4088-4104.
doi: 10.1016/j.na.2011.03.043. |
[6] |
J. Huang and Y. Zhao, Extended quasi-homogeneous polynomial system in $\mathbb{R}^3$, submitted. |
[7] |
M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. |
[8] |
R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253. |
[9] |
J. Llibre, C. A. Buzzi and P. R. da Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst., 17 (2007), 529-540. |
[10] |
J. Llibre, J. S. Perez Del Rio and J. A. Rodriguez, Structural stability of planar homogeneous polynomial vector fields: applications to critical points and to infinity, Journal of Differential Equations, 125 (1996), 490-520.
doi: 10.1006/jdeq.1996.0038. |
[11] |
J. Llibre and C. Pessoa, Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere, Rend. Circ. Mat. Palermo, 55 (2006), 63-81.
doi: 10.1007/BF02874668. |
[12] |
J. Llibre and C. Pessoa, homogeneous polynomial vector fields of degree $2$ on the $2$-dimensional sphere, Extracta Math., 21 (2006), 167-190. |
[13] |
J. Llibre and H. Wu, Hopf bifurcation for degenerate singular points of multiplicity $2n-1$ in dimension $3$, Bull. Sci. Math., 132 (2008), 218-231.
doi: 10.1016/j.bulsci.2007.01.003. |
[14] |
J. Llibre and J. Yu, Limit cycles for a class of three-dimensional polynomial differential systems, Journal of Dynamical and Control Systems, 13 (2007), 531-539.
doi: 10.1007/s10883-007-9025-5. |
[15] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[16] |
J. Llibre and X. Zhang, Hopf bifurcation in higher dimensional differential systems via the averaging method, Pacific J. Math., 240 (2009), 321-341.
doi: 10.2140/pjm.2009.240.321. |
[17] |
L. Markus, Quadratic differential equations and non-associative algebras, in "Contributions to the Theory of Non-linear Oscillations" (ed. S. Lfschetz), Princeton, Univ. Press, (1960), 185-213. |
[18] |
Y. Ye et al., "Theory of Limit Cycles," Transl. Math. Monogr. American Mathematical Society, Providence, 1986. |
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