Effect of coupling, synchronization of chaos and stick-slip motion in two mutually coupled dynamical systems

In this work, we study the synchronization of two coupled chaotic oscillators. The uncoupled system corresponds to a mass attached to a nonlinear spring and driven by a rolling carpet. For identical oscillators, complete synchronization is analyzed using Lyapunov stability theory. This first analysis reveals that stability area of synchronization increases with the values of the coupling coefficient. Numerical simulations are shown to illustrate and validate stick-slip and chaos synchronization. Some cases of anti-synchronization are detected. Curiously, amplification of fixed point either regular or chaotic is observed in the area of anti-synchronization. Furthermore, phase synchronization is studied for nonidentical oscillators. It appears that for certain values of the coupling coefficient, coincidence of the phases is obtained, while the amplitudes remain uncorrelated. Contrarily to the case of complete synchronization, it does not exist a threshold of the coupling from which phase synchronization could appear. Besides, when we add the modified tuned mass damper on the structure, the behavior of the system can change including the appearance of synchronization, particularly in the region of fixed point. More precisely, complete synchronization is improved in the region of fixed point, while the damage of synchronization is observed when the velocity of the carpets is less than \(0.30\).