On the periodic solutions of a class of Duffing differential equations

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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

Abstract
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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 - A, 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.
[2]	H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations,, Nonlinearity, 21 (2008), 2485.
[3]	G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung,, Sammlung Viewg Heft, 41/42 (1918).
[4]	J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation,, in:, 9 (1997), 115.
[5]	R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type,, Boo. Uni. Mat. Ital B, 3 (1989), 533.
[6]	F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'', Universitext, (1991).

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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.
[2]	H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations,, Nonlinearity, 21 (2008), 2485.
[3]	G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung,, Sammlung Viewg Heft, 41/42 (1918).
[4]	J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation,, in:, 9 (1997), 115.
[5]	R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type,, Boo. Uni. Mat. Ital B, 3 (1989), 533.
[6]	F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'', Universitext, (1991).

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