Bifurcation and stability properties of periodic solutions to two nonlinear spring-mass systems

The behavior of a periodically forced, linearly damped mass suspended by a linear spring is well known. In this paper we study the nature of periodic solutions to two nonlinear spring-mass equations; our nonlinear terms are similar to earlier models of motion in suspension bridges. We contrast the multiplicity, bifurcation, and stability of periodic solutions for a piecewise linear and smooth nonlinear restoring force. We find that while many of the qualitative properties are the same for the two models, the nature of the secondary bifurcations (period-doubling and quadrupling) differs significantly.