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On the periodic solutions of a class of Duffing differential equations

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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.
    Mathematics Subject Classification: Primary: 34C23, 34C25, 34C29, 34D20, 34G15.


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  • [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.


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


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


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


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


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

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