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

Abstract

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

A non-autonomous kind of Duffing equation

“…For more information on the history of these differential equations see the paper of Mawhin [13]. The almost 200 references quoted in this last paper are on the periodic solutions of different kind of Duffing equations, and from its publication many new papers working on these type of periodic solutions also have been published, see the papers [3,4,17] and the quoted references therein. See also the non-autonomous differential equation studied in [7].…”

Reversed S-Shaped Bifurcation Curve for a Neumann Problem

“…Chen and Li [7] also studied the Duffing equation (2) in a very particular case, i.e., ( ) ≡ > 0, ( ) = 1, and , constants. They obtained that (2) has exactly three T-periodic solutions.…”

Minimization of eigenvalues and construction of non-degenerate potentials for the one-dimensional p-Laplacian

“…The solvability, the precise multiplicity and the stability of solutions of several interesting classes of superlinear differential equations of the Landesman-Lazer type or of the Ambrosetti-Prodi type also can be obtained from the characterization of non-degenerate potentials [7,8,11,17,22]. However, to give an explicit construction for non-degenerate potentials is a difficult problem.…”