Saddle-node bifurcations on approximate inertial manifolds for a driven wave equation

Abstract

A strongly damped and driven nonlinear wave equation describing nonlinear vibrations of axially moving beams is investigated. We establish the existence of an approximate inertial manifold and obtain the amplitude frequency response of the reduced system on this manifold by using multiscale method. The numerical calculations show that, as forcing frequency and damping coefficient are varied, the saddle-node bifurcations can be detected in the two-parameter plane. Thus, an arbitrary small perturbation may lead to a sudden amplitude jump corresponding to a relatively large amplitude response for the primary resonance.

Keywords:

• saddle-node bifurcation
• approximate inertial manifold
• multiscale method
• nonlinear wave equation

AMS Subject Classifications:

• 58J55
• 74G60
• 35B42

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the support of the National Natural Science Foundation of China [grant number 11471281] and [grant number 11372102] for the work reported in this paper.