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July  2013, 33(7): 2861-2883. doi: 10.3934/dcds.2013.33.2861

## 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

Received  March 2012 Revised  June 2012 Published  January 2013

In this paper, we study the bifurcation of isolated closed orbits from degenerated singularity of $3$-dimensional polynomial system $dx/dt = Q(x)$. For some types of $Q(x)$, we get the lower bound for the number of these isolated closed orbits. In particular cases, an explicit (sometimes sharp) upper bound is obtained. Using these results, we investigate degenerated Hopf bifurcation and give a sufficient condition for the existence of isolated closed orbits. Also we show that the $3$ species model of degree $3$ admits $2$ isolated closed orbits bifurcating from origin.
Citation: Jianfeng Huang, Yulin Zhao. Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2861-2883. doi: 10.3934/dcds.2013.33.2861
##### 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.  Google Scholar [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.  Google Scholar [3] R. E. Gomory, Trajectories tending to a critical point in 3-space, Annals of Mathematics, 61 (1955), 140-153.  Google Scholar [4] J. Huang and Y. Zhao, The limit set of trajectory in quasi-homogeneous system on $\mathbbR^3$,, Applicable Analysis., ().  doi: 10.1080/00036811.2011.567193.  Google Scholar [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.  Google Scholar [6] J. Huang and Y. Zhao, Extended quasi-homogeneous polynomial system in $\mathbbR^3$,, submitted., ().   Google Scholar [7] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71.  Google Scholar [8] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Llibre and C. Pessoa, homogeneous polynomial vector fields of degree $2$ on the $2$-dimensional sphere, Extracta Math., 21 (2006), 167-190.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Y. Ye et al., "Theory of Limit Cycles," Transl. Math. Monogr. American Mathematical Society, Providence, 1986.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] R. E. Gomory, Trajectories tending to a critical point in 3-space, Annals of Mathematics, 61 (1955), 140-153.  Google Scholar [4] J. Huang and Y. Zhao, The limit set of trajectory in quasi-homogeneous system on $\mathbbR^3$,, Applicable Analysis., ().  doi: 10.1080/00036811.2011.567193.  Google Scholar [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.  Google Scholar [6] J. Huang and Y. Zhao, Extended quasi-homogeneous polynomial system in $\mathbbR^3$,, submitted., ().   Google Scholar [7] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71.  Google Scholar [8] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), 243-253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Llibre and C. Pessoa, homogeneous polynomial vector fields of degree $2$ on the $2$-dimensional sphere, Extracta Math., 21 (2006), 167-190.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] Y. Ye et al., "Theory of Limit Cycles," Transl. Math. Monogr. American Mathematical Society, Providence, 1986.  Google Scholar
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