Bifurcation of isolated closed orbits from degenerated singularity in \mathbb{R}^{3}

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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 $\mathbb{R}^{3}$

Jianfeng Huang ^1, and Yulin Zhao ^2,

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

Abstract
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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.

Keywords: three-dimensional system., isolated closed orbits, Bifurcation, degenerated singularity, quasi-homogeneous vector fields.

Mathematics Subject Classification: 37C10, 37C27, 37G10, 37G1.

Citation: Jianfeng Huang, Yulin Zhao. Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$. Discrete and 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.
[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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