# American Institute of Mathematical Sciences

March  2014, 19(2): 485-522. doi: 10.3934/dcdsb.2014.19.485

## Identification of focus and center in a 3-dimensional system

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China, China, China 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  May 2013 Revised  October 2013 Published  February 2014

In this paper we identify focus and center for a generalized Lorenz system, a 3-dimensional quadratic polynomial differential system with four parameters $a$, $b$, $c$, $\sigma$. The known work computes the first order Lyapunov quantity on a center manifold and shows the appearance of a limit cycle for $a\neq b$, but the order of weak foci was not determined yet. Moreover, the case that $a=b$ was not discussed. In this paper, for $a\neq b$ we use resultants to decompose the algebraic varieties of Lyapunov quantities so as to prove that the order is at most 3. Further, we apply Sturm's theorem to determine real zeros of the first order Lyapunov quantity over an extension field so that we give branches of parameter curves for each order of weak foci. For $a=b$ we prove its Darboux integrability by finding an invariant surface, showing that the equilibrium of center-focus type is actually a rough center on a center manifold.
Citation: Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485
##### References:
 [1] M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Springer, Berlin, 2004. [2] D. V. Anosov and V. I. Arnold, Dynamical systems I. Ordinary differential equations and smooth dynamical systems, Translated from the Russian, Springer, Berlin, 1988. doi: 10.1007/978-3-642-61551-1. [3] J. Bak and D. J. Newman, Complex Analysis, Springer, New York, 2010. doi: 10.1007/978-1-4419-7288-0. [4] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation, 1954 (1954), no. 100, 19pp. [5] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981. [6] X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581. doi: 10.1016/j.cam.2009.06.029. [7] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312 (1989), 433-486. doi: 10.1090/S0002-9947-1989-0930075-2. [8] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251. Springer-Verlag, New York-Berlin, 1982. [9] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), (French) Bull. Sci. Math., 2 (1878), 60-96. [10] M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347. [11] J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger, New York, 1980. [12] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1995. [13] T. Li, G. Chen, Y. Tang and L. Yang, Hopf bifurcation of the generalized Lorenz canonical form, Nonlinear Dynam., 47 (2007), 367-375. doi: 10.1007/s11071-006-9036-x. [14] J. Llibre, C. A. Buzzi and P. R. Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst., 17 (2007), 529-540. [15] A. M. Lyapunov, Stability of Motions, Academic Press, New York, 1966. doi: 10.1080/00207179208934253. [16] L. F. Mello and S. F. Coelho, Degenerate Hopf bifurcations in the Lšystem, Phys. Lett. A, 373 (2009), 1116-1120. doi: 10.1016/j.physleta.2009.01.049. [17] B. Mishra, Algorithmic Algebra, Springer, New York, 1993. [18] W. Miller, Symmetry Groups and Their Applications, Academic Press, New York, 1972. [19] H. Poincaré, Mémoire sur les courbes définies par une équation difféentielle, (French) J. Math. Pure Appl., 1 (1881), 375-422. [20] A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer, Berlin, 2001. [21] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Boston-Basel-Berlin: Birkhäuser, 2009. doi: 10.1007/978-0-8176-4727-8. [22] W. Zhang, X. Hou and Z. Zeng, Weak center and bifurcation of critical periods in reversible cubic systems, Comput. Math. Appl., 40 (2000), 771-782. doi: 10.1016/S0898-1221(00)00195-4. [23] H. Zoladek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860. doi: 10.1088/0951-7715/8/5/011.

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##### References:
 [1] M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Springer, Berlin, 2004. [2] D. V. Anosov and V. I. Arnold, Dynamical systems I. Ordinary differential equations and smooth dynamical systems, Translated from the Russian, Springer, Berlin, 1988. doi: 10.1007/978-3-642-61551-1. [3] J. Bak and D. J. Newman, Complex Analysis, Springer, New York, 2010. doi: 10.1007/978-1-4419-7288-0. [4] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation, 1954 (1954), no. 100, 19pp. [5] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981. [6] X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581. doi: 10.1016/j.cam.2009.06.029. [7] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312 (1989), 433-486. doi: 10.1090/S0002-9947-1989-0930075-2. [8] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 251. Springer-Verlag, New York-Berlin, 1982. [9] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), (French) Bull. Sci. Math., 2 (1878), 60-96. [10] M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347. [11] J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger, New York, 1980. [12] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1995. [13] T. Li, G. Chen, Y. Tang and L. Yang, Hopf bifurcation of the generalized Lorenz canonical form, Nonlinear Dynam., 47 (2007), 367-375. doi: 10.1007/s11071-006-9036-x. [14] J. Llibre, C. A. Buzzi and P. R. Silva, 3-dimensional Hopf bifurcation via averaging theory, Discrete Contin. Dyn. Syst., 17 (2007), 529-540. [15] A. M. Lyapunov, Stability of Motions, Academic Press, New York, 1966. doi: 10.1080/00207179208934253. [16] L. F. Mello and S. F. Coelho, Degenerate Hopf bifurcations in the Lšystem, Phys. Lett. A, 373 (2009), 1116-1120. doi: 10.1016/j.physleta.2009.01.049. [17] B. Mishra, Algorithmic Algebra, Springer, New York, 1993. [18] W. Miller, Symmetry Groups and Their Applications, Academic Press, New York, 1972. [19] H. Poincaré, Mémoire sur les courbes définies par une équation difféentielle, (French) J. Math. Pure Appl., 1 (1881), 375-422. [20] A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer, Berlin, 2001. [21] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Boston-Basel-Berlin: Birkhäuser, 2009. doi: 10.1007/978-0-8176-4727-8. [22] W. Zhang, X. Hou and Z. Zeng, Weak center and bifurcation of critical periods in reversible cubic systems, Comput. Math. Appl., 40 (2000), 771-782. doi: 10.1016/S0898-1221(00)00195-4. [23] H. Zoladek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860. doi: 10.1088/0951-7715/8/5/011.
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