Numerical Analysis of Oscillation Death in Coupled Self-Excited Elastic Beams

The emergence of the oscillation death phenomenon in a ring of four coupled self-excited elastic beams is numerically explored in this work. The beams are mathematically represented through partial differential equations which are solved by means of the finite differences method. A coupling scheme based on shared boundary conditions at the roots of the beams is assumed, and as initial conditions, zero velocity of the first beam and three normal vibration modes of a linear elastic beam are employed. The influence of the self-exciting constant on the ring dynamics is analyzed. It is observed that oscillation death arises as result of the singularity of the coupling matrix. 1. Introduction In the past years the collective behavior of coupled nonlinear oscillators has been widely studied in many disciplines, for example, physics [1], biology [2], ecology [3], chemistry [4], and mechanics [5]. A wide diversity of nonlinear dynamic phenomena such as locking [1], partial synchronization [6], full synchronization [7], antiphase synchronization [8], and clustering [9] have been reported in coupled oscillators. Many coupling schemes have also been tested: local [10], nearest [11], global [12], diffusive [13], adaptive [14], delayed [15], hierarchical [16], and so on. An interesting behavior of coupled oscillators is amplitude death and oscillation death, which are steady states where the coupled oscillators stop their oscillation in a permanent way and become frozen in time [17–19]. Sometimes this cessation of oscillations in time is named quenching [20]. Amplitude death arises through a Hopf bifurcation mechanism in coupled oscillators with an important parameter mismatch or in identical oscillators with time delays [21]. An already existing unstable steady state with zero amplitude is transformed by the coupling into a stable one allowing its observation; that is, the coupling induces stability at the origin of the phase space. On the other hand, oscillation death occurs through a saddle-node bifurcation mechanism allowing the emergence of new fixed points: a new stable steady state with nonzero amplitude is created by the coupling [19, 21]. Frequently, in the literature amplitude death is confused with oscillation death [22–27]. Even the famous finding of Lord Rayleigh [28] related to the quenching of two organ pipes standing side by side is indistinctly considered as amplitude death or oscillation death [29]. To date, in spite of the significant conceptual and technical differences between amplitude death and oscillation death, there is not yet a clear