Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

Chen, Hongbin; Li, Yi

doi:10.1090/S0002-9939-07-09024-7

Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities
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Proc. Amer. Math. Soc. 135 (2007), 3925-3932 Request permission

Abstract:

We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities, \begin{equation*} x''+cx’+ax-x^{3}=h(t),\tag {$*$} \end{equation*} where $a$ and $c>0$ are positive constants and $h(t)$ is a positive $T$-periodic function. We obtain sharp bounds for $h$ such that $(*)$ has exactly three ordered $T$-periodic solutions. Moreover, when $h$ is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.

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