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January  2013, 33(1): 277-282. doi: 10.3934/dcds.2013.33.277

On the periodic solutions of a class of Duffing differential equations

Jaume Llibre 1, and  Luci Any Roberto 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2. 

Departamento de Matemática, Ibilce -UNESP, 15054-000 São José do Rio Preto, Brazil

Received  February 2011 Revised  December 2011 Published  September 2012

In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$, $\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions and $\epsilon$ is sufficiently small. Our results are proved using the averaging method of first order.
Keywords: bifurcation, stability., Duffing differential equation, averaging method, Periodic solution.
Mathematics Subject Classification: Primary: 34C23, 34C25, 34C29, 34D20, 34G1.
Citation: Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
References:
[1]

H. B. Chen and Y. Li, Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities, Proc. Amer. Math. Soc., 135 (2007), 3925-3932. Google Scholar

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 21 (2008), 2485-2503. Google Scholar

[3]

G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung, Sammlung Viewg Heft, Viewg, Braunschweig, 41/42 (1918). Google Scholar

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J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation, in: "Equadiff, Brno. Proceedings, Brno: Masaryk University,'' 9 (1997), 115-145. Google Scholar

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R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type, Boo. Uni. Mat. Ital B, 3 (1989), 533-546.  Google Scholar

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F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'' Universitext, Springer, 1991.  Google Scholar

show all references

References:
[1]

H. B. Chen and Y. Li, Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities, Proc. Amer. Math. Soc., 135 (2007), 3925-3932. Google Scholar

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 21 (2008), 2485-2503. Google Scholar

[3]

G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung, Sammlung Viewg Heft, Viewg, Braunschweig, 41/42 (1918). Google Scholar

[4]

J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation, in: "Equadiff, Brno. Proceedings, Brno: Masaryk University,'' 9 (1997), 115-145. Google Scholar

[5]

R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type, Boo. Uni. Mat. Ital B, 3 (1989), 533-546.  Google Scholar

[6]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'' Universitext, Springer, 1991.  Google Scholar

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