Bifurcation and stability of periodic solutions of Duffing equations

Abstract

We study the stability and exact multiplicity of periodic solutions of the Duffing equation from the global bifurcation point of view and show that the Duffing equation with cubic nonlinearities has at most three T -periodic solutions under a strong damped condition. More precisely, we prove that the T -periodic solutions form a smooth S-shaped curve and the stability of each T -periodic solution is determined by Floquet theory.

On Stability of Periodic Solutions of Lienard Type Equations

“…When ( ) and ( ) are continuous, the conclusion of Theorem 2 still holds if the second and third term in (8) or (9) are replaced by…”

“…we have the following lemma which is given by the author in [9] Lemma 10. Let ( ) ∈ ∞ (0, ) such that…”

“…In [7], Lazer and Mckenna established stability results by converting the equation to a fixed point problem. Recently, more complete results concerning the stability and the sharpness of the rate of decay of periodic solutions were obtained by Chen and Li in [8,9].…”

A note on the periodic orbits of a kind of Duffing equations

“…For more details on the history of these differential equations see the paper of Mawhin [9]. Many of the 190 references quoted in this last paper are on the periodic orbits of different kind of Duffing equations, and from its publication many new papers working on these type of periodic orbits also have been published, see for instance the papers [3,4,12] and the quoted references in there.…”

“…DOI: [10.1016/j.amc.2012.11.087] Theorem 1 will be proved in section 3 using the averaging theory, see section 2 where is described the result on averaging theory that we shall need to use here. In general the main difficulty for studying the existence of periodic orbits using the averaging theory is to find a change of variables which write the studied differential system into the normal form of the averaging theory (see (4)). Now we shall do some applications of Theorem 1.…”