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

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ISSN 1088-6826(online) ISSN 0002-9939(print)

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

Authors: Hongbin Chen and Yi Li
Journal: Proc. Amer. Math. Soc. 135 (2007), 3925-3932
MSC (2000): Primary 34C10, 34C25
Posted: September 7, 2007
MathSciNet review: 2341942
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities,

$\displaystyle x''+cx'+ax-x^{3}=h(t),\tag{$*$}$

where

and

are positive constants and

is a positive

-periodic function. We obtain sharp bounds for

such that

has exactly three ordered

-periodic solutions. Moreover, when

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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