The dynamics of the two nonlinearly coupled oscillators

In this paper the dynamics of two elastically coupled pendulums is studied. The pendulums oscillate under the influence of external rotational moments, their masses are considered to be equal. The current work is motivated by multiple applications in physics and biology that the model has. Due to the fact that most of the previous studies focused on similar systems of higher order, we believe that the current research can serve as a basis for understanding the functioning of more complex oscillatory ensembles. It is, therefore, vital to provide a complete study of the system dynamics for different parameter values. Throughout the study different regimes of the system activity are examined. Thus, non-oscillatory mode, synchronization, periodic and quasi-periodic regimes are discussed in the paper. Synchronization is often considered to be one of the most important forms of interaction between oscillatory elements of various nature. For this reason the synchronization domain is thoroughly investigated in this paper. The main results of the current research are as follow. An analytical approximation of the synchronization domain border is obtained in $(d,\alpha)$ parameter plane. Here $d$ denotes the coupling strength, whereas $\alpha$ is the synchronization parameter. By means of numerical integration methods the approximation is also shown to be accurate. In order to provide better understanding of the regimes that exist in the system for various parameter values, bifurcation diagrams for several values of the coupling parameter in $(\gamma_1, \gamma_2)$ plane are drawn.