August  2014, 19(6): 1731-1736. doi: 10.3934/dcdsb.2014.19.1731

Zero-Hopf bifurcation for a class of Lorenz-type systems

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340

Received  May 2013 Revised  February 2014 Published  June 2014

In this paper we apply the averaging theory to a class of three-dimensional autonomous quadratic polynomial differential systems of Lorenz-type, to show the existence of limit cycles bifurcating from a degenerate zero-Hopf equilibrium.
Citation: Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731
References:
[1]

A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree,, Bull. Sci. Math., 128 (2004), 7. doi: 10.1016/j.bulsci.2003.09.002. Google Scholar

[2]

J. Guckenheimer, On a codimension two bifurcation,, Lecture Notes in Math., 898 (1980), 99. Google Scholar

[3]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, Springer, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar

[4]

M. Han, Existence of periodic orbits and invariant tori in codimension two bifurcations of three dimensional systems,, J. Sys. Sci $&$ Math. Scis., 18 (1998), 403. Google Scholar

[5]

C. Hua, G. Chen, Q. Li and J. Ge, Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system,, Discrete and Continuous Dynamical Systems, 16 (2011), 475. doi: 10.3934/dcdsb.2011.16.475. Google Scholar

[6]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-3978-7. Google Scholar

[7]

J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations,, Mathematical Proceedings of the Cambridge Philosophical Society, 148 (2010), 363. doi: 10.1017/S0305004109990193. Google Scholar

[8]

J. Llibre and C. Valls, Hopf bifurcation for some analytic differential systems in $R^3$ via averaging theory,, Discrete, 30 (2011), 779. doi: 10.3934/dcds.2011.30.779. Google Scholar

[9]

J. Llibre and X. Zhang, Hopf bifurcation in higher dimensional differential systems via the averaging method,, Pacific J. Math., 240 (2009), 321. doi: 10.2140/pjm.2009.240.321. Google Scholar

[10]

D. H. Pi and X. Zhang, Limit cycles of differential systems via the averaging methods,, Can. Appl. Math. Q., 17 (2009), 243. Google Scholar

[11]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems,, Applied Mathematical Sci., 59 (1985). doi: 10.1007/978-1-4757-4575-7. Google Scholar

[12]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar

show all references

References:
[1]

A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree,, Bull. Sci. Math., 128 (2004), 7. doi: 10.1016/j.bulsci.2003.09.002. Google Scholar

[2]

J. Guckenheimer, On a codimension two bifurcation,, Lecture Notes in Math., 898 (1980), 99. Google Scholar

[3]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,, Springer, (1983). doi: 10.1007/978-1-4612-1140-2. Google Scholar

[4]

M. Han, Existence of periodic orbits and invariant tori in codimension two bifurcations of three dimensional systems,, J. Sys. Sci $&$ Math. Scis., 18 (1998), 403. Google Scholar

[5]

C. Hua, G. Chen, Q. Li and J. Ge, Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system,, Discrete and Continuous Dynamical Systems, 16 (2011), 475. doi: 10.3934/dcdsb.2011.16.475. Google Scholar

[6]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-3978-7. Google Scholar

[7]

J. Llibre, A. C. Mereu and M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations,, Mathematical Proceedings of the Cambridge Philosophical Society, 148 (2010), 363. doi: 10.1017/S0305004109990193. Google Scholar

[8]

J. Llibre and C. Valls, Hopf bifurcation for some analytic differential systems in $R^3$ via averaging theory,, Discrete, 30 (2011), 779. doi: 10.3934/dcds.2011.30.779. Google Scholar

[9]

J. Llibre and X. Zhang, Hopf bifurcation in higher dimensional differential systems via the averaging method,, Pacific J. Math., 240 (2009), 321. doi: 10.2140/pjm.2009.240.321. Google Scholar

[10]

D. H. Pi and X. Zhang, Limit cycles of differential systems via the averaging methods,, Can. Appl. Math. Q., 17 (2009), 243. Google Scholar

[11]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems,, Applied Mathematical Sci., 59 (1985). doi: 10.1007/978-1-4757-4575-7. Google Scholar

[12]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Universitext, (1996). doi: 10.1007/978-3-642-61453-8. Google Scholar

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