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Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold
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 |
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] | |
[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. |
show all references
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] | |
[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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