Nonlinear Dyn (2010) 59: 711–731 DOI 10.1007/s11071-009-9572-2 ORIGINAL PAPER Bifurcations of attractors in forced system with nonlinear energy sink: the effect of mass asymmetry Y. Starosvetsky · O.V. Gendelman Received: 10 May 2009 / Accepted: 26 July 2009 / Published online: 11 September 2009 © Springer Science+Business Media B.V. 2009 Abstract System under investigation comprises a har- monically forced linear oscillator and a nonlinear en- ergy sink (NES). The NES is a small mass (rel- ative to that of the linear oscillator) which is at- tached to the primary system via a linear damper and strongly nonlinear spring (pure cubic nonlinearity). Among possible responses there exists one charac- terized by extremely deep modulation of the oscilla- tions and referred to as a strongly modulated response regime (SMR). Numeric simulations demonstrate that the SMR can exist only for sufficiently small values of the NES mass. Known analytical approximations for description of the SMR deal with the lowest order of the asymptotic approximation and, consequently, work fairly well only for very small values of the NES mass and do not take into account its actual value. In the present study, we develop the analytical tools to in- vestigate the higher-order asymptotic approximation. This enables us to depict the qualitative changes in the regime for the growing values of a NES mass and also to provide a crude estimation for a NES mass threshold. It is also demonstrated that in some cases the mechanisms of loss of stability by SMR (due to the growing values of NES mass) can be illustrated and explained via one-dimensional mapping diagrams. Y. Starosvetsky · O.V. Gendelman ( ) Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 32000 Israel e-mail: ovgend@tx.technion.ac.il The described novel analytical approach is verified nu- merically and a fairly good agreement between the nu- merical and analytical models is observed. Keywords Nonlinear energy sink · Attractor · Relaxation oscillations · Global bifurcations Introduction Vibration suppression in various engineering devices is a one of the extensively studied topics over the last three decades. One of the well-known solutions in vi- bration mitigation is the introduction of a weakly non- linear vibration absorber. This solution is quite simple for realization and is based on the addition of a weakly nonlinear degree of freedom to a main system given to various types of external excitation [1–11]. The ef- fective bandwidth is governed by the damping in the absorber and a trade-off exists between attenuation ef- ficiency and bandwidth. Applications of nonlinear energy sink (NES) for the enhanced vibration mitigation in various engineering devices is one of the intensively studied topics of the last decade. Recent studies of the systems subject to external loading and comprising NES (attached to the main system) have demonstrated outstanding capabil- ities of the NES in a sense of vibration suppression. By the term NES we refer to a small mass (relative to that of the linear oscillator) which is attached to the primer via a linear damper and strongly nonlinear