Multistability and basin stability in coupled pendulum clocks.

Abstract

In this paper, we investigate the phenomenon of multistability and the concept of basin stability in two coupled pendula with escapement mechanisms, suspended on horizontally oscillating beam. The dynamics of a single pendulum clock is studied and described, showing possible responses of the unit. The basin stability maps are discussed in two-parameters plane, where we vary both the system's stiffness as well as the damping. The possible attractors for the investigated clocks are discussed, showing that different patterns of synchronization and desynchronization can occur. The oscillators may completely synchronize in one of the three possible combinations (including inphase and antiphase ones), practically synchronize with some fluctuations or stay in the irregular pattern, which includes chaotic motion. The transitions between solutions are studied, uncovering that the road from one type of dynamics into another may become very complex. Moreover, we examine the multistability property of our model using the bifurcation diagrams and the basins of attraction maps, discussing possible scenarios in which the states co-exist. The analysis of attractors' basins uncovers complicated structure of the latter ones, exhibiting that the final behavior of investigated model may be hard to determine and trace. Our results are discussed for the cases of identical and nonidentical pendula, as well as light and heavy beam, showing that depending on considered scenario, various patterns of behaviors and transitions may be observed. The research described in this paper proves that the mechanical properties of the system's suspension may play a crucial role in the possibility of the appearance of different types of attractors and that the basin stabilities of states strictly depend on the values of considered parameters.