Non–continuation of the periodic oscillations of a forced pendulum in the presence of friction

Abstract

Abstract. A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.

Degenerate Equations of Pendulum-Type

“…(2.1) with c = 0 can be degenerate only whenf = 0. The situation is different when c > 0 and the existence of periodic solution is not guaranteed byf = 0 (see [22]). Now, the given parametrization allows us to prove a complementary result: when c > 0 there exist degenerate equations for whichf = 0.…”

Periodic solutions of resonant systems with rapidly rotating nonlinearities

“…From the discussion following Theorem 1.1 it is immediately seen that, for a given p, the condition This means that, in contrast with (2), oscillations of g around 0 at ±∞ are, in fact, allowed, but the length of the intervals I ± where g does not change sign is determined by g and p, and cannot be arbitrarily small. For instance, when g(u) = sin u, there are well known examples of forcing terms p with zero average and c = 0 such that the problem has no solutions (see [1], [7], [9]).…”