American Institute of Mathematical Sciences

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

Identification of focus and center in a 3-dimensional system

Lingling Liu 1, , Bo Gao 1, , Dongmei Xiao 2, and  Weinian Zhang 1,

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.
Keywords: center-focus, Darboux integrability., algebraic variety, Generalized Lorenz system, extension field.
Mathematics Subject Classification: 34C05, 34C07, 34C4.
Citation: Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

J. Bak and D. J. Newman, Complex Analysis, Springer, New York, 2010. doi: 10.1007/978-1-4419-7288-0.  Google Scholar

[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.  Google Scholar

[5]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), (French) Bull. Sci. Math., 2 (1878), 60-96. Google Scholar

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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.  Google Scholar

[11]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger, New York, 1980.  Google Scholar

[12]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. M. Lyapunov, Stability of Motions, Academic Press, New York, 1966. doi: 10.1080/00207179208934253.  Google Scholar

[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.  Google Scholar

[17]

B. Mishra, Algorithmic Algebra, Springer, New York, 1993.  Google Scholar

[18]

W. Miller, Symmetry Groups and Their Applications, Academic Press, New York, 1972.  Google Scholar

[19]

H. Poincaré, Mémoire sur les courbes définies par une équation difféentielle, (French) J. Math. Pure Appl., 1 (1881), 375-422. Google Scholar

[20]

A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Adler, P. van Moerbeke and P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Springer, Berlin, 2004.  Google Scholar

[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.  Google Scholar

[3]

J. Bak and D. J. Newman, Complex Analysis, Springer, New York, 2010. doi: 10.1007/978-1-4419-7288-0.  Google Scholar

[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.  Google Scholar

[5]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger, New York, 1980.  Google Scholar

[12]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

A. M. Lyapunov, Stability of Motions, Academic Press, New York, 1966. doi: 10.1080/00207179208934253.  Google Scholar

[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.  Google Scholar

[17]

B. Mishra, Algorithmic Algebra, Springer, New York, 1993.  Google Scholar

[18]

W. Miller, Symmetry Groups and Their Applications, Academic Press, New York, 1972.  Google Scholar

[19]

H. Poincaré, Mémoire sur les courbes définies par une équation difféentielle, (French) J. Math. Pure Appl., 1 (1881), 375-422. Google Scholar

[20]

A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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