On the periodic solutions of a class of Duffing differential equations

Abstract

We provide sufficient conditions for the existence of periodic solutions in the class of Duffing differential equationswhere the functions a(t), b(t), c(t) and h(t, x) are C 2 and T -periodic in the variable t.

Stable quasi-periodic orbits of a class of quintic Duffing systems

“…Early uses of this theory go back to Bogolyubov's averaging principle, originally described in [5]. Thus we get the following assertion, which generalizes the results exposed in [11].…”

“…We generalize the results in [11], from a cubic to a quintic Duffing equation without cubic nonlinearity,…”

“…where cubic nonlinearity appears in the approximation of the sine function, here h(t) = h(t + 2π) (see, for instance, [10]). In [11] (see also [3]), averaging methods are used to show existence and local stability of some generalizations for periodic time-dependent coefficients,…”

The rate of decay of stable periodic solutions for Duffing equation with L p -conditions

“…The existence and multiplicity of T -periodic solutions of (1.1) and more general equations have drawn a lot of attentions in the past three decades [9,12,13,15,[18][19][20]25,29,31,40]. For more details, we refer to the survey of Mawhin [22].…”

“…Similar ideas have been posed several times. For example, Llibre and Roberto [19], Torres and Zhang [30,32], Wang and Li [34] investigate the multiplicity and stability of T -periodic solutions of second order equation, with L p -conditions for g x (t, x). Zhang and Li [36,38,39] develop some new estimates of the periodic and anti-periodic eigenvalues for the Hill's equation…”