Manifolds of Nonlinear Dynamic Single-DOF Systems

Abstract

This paper deals with the long term behaviour (attractors) of nonlinear dynamic single degree of freedom (DOF) systems, excited by a periodic external load. Different attractors can exist for one set of system-parameters. The set of initial conditions of trajectories which approach one attractor is called the basin of attraction of the attractor. The boundaries of the basins of attraction are formed by the stable manifolds of unstable periodic solutions. These stable manifolds are the set of initial conditions of trajectories which approach an unstable period solution (saddle). Because these are the only trajectories which do not approach an attractor, in general the stable manifolds are the boundaries of the basins of attraction. When stable and unstable manifolds intersect, a chaotic attractor or fractal boundaries of basins of attraction are created. These phenomena are demonstrated by calculating the manifolds of two single-DOF systems, one with a cubic stiffening spring and one with an one-sided spring.