Bifurcation analysis of non-linear oscillators interacting via soft impacts

In this paper we present a bifurcation analysis of two periodically forced Duffing oscillators coupled via soft impact. The controlling parameters are the distance between the oscillators and the difference in the phase of the harmonic excitation. In our previous paper http://arXiv:1602.04214 (P. Brzeski et al. Controlling multistability in coupled systems with soft impacts [11]) we show that in the multistable system we are able to change the number of stable attractors and reduce the number of co-existing solutions via transient impacts. Now we perform a detailed path-following analysis to show the sequence of bifurcations which cause the destabilization of solutions when we decrease the distance between the oscillating systems. Our analysis shows that all solutions lose stability via grazing-induced bifurcations (period doubling, fold or torus bifurcations). The obtained results provide a deeper understanding of the mechanism of reduction of the multistability and confirmed that by adjusting the coupling parameters we are able to control the system dynamics.