The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction
This paper deals with an equation of pendulum type having friction and periodic forcing. When the unknown is interpreted as an angle, the equation becomes a dissipative system on the cylinder. In this situation the attractor, understood as the set of bounded solutions, is compact and nonempty. It is known, that this attractor is homeomorphic to a circle (Jordan curve) when the friction is large. In this paper, the author obtains a complementary result: the existence of an inversely unstable periodic solution implies that the attractor is not a Jordan curve. The notion of inversely unstable periodic solution was introduced by Levinson and had an analytical character. Namely, this notion referred to a hyperbolic solution with negative Floquet multipliers. The new notion introduced in this paper is based on topological degree and extends Levinson's definition.
scientific article; zbMATH DE number 2173048