# Double Cross (2020) 01.mp4

In Fig. 5a, one can observe a sample of the time plot, which has been calculated for the system with coefficients \(A = 3.5 \, [\text mm], f = 4.4 \, [\text Hz], k_1 = 0.033 \, [\text N m], k_2 = 0.053 \, [\text N m]\). The excitation parameters correspond to the case of both oscillations and rotations of the upper and lower pendula bobs. The evolution of the difference between the displacements of the lower bobs \((\varphi _2^1-\varphi _2^2)\) marked in the vertical axis in Fig. 5a exhibits that after some transient, random movements, the oscillators begin to synchronize. The coherent state is reached around \(t \approx 250 \, T \, [\text s]\) and has been observed also for the upper pendula bobs (not shown in the figure). However, at some critical point (\(t \approx 440 \, T \, [\text s]\)), the oscillators desynchronize and the common chaotic behavior is lost. To investigate the mechanism of the synchronization disappearance, we have zoomed the time plots around the critical point (\(t \approx 440 \, T \, [\text s]\)), which is shown in Fig. 5b. The displacements of the first and the second lower pendula bobs are marked in blue and cyan, respectively. As can be seen from the very beginning in Fig. 5b, the rotations of both pendula induce small differences in their movements, which is caused by the hyperchaotic character of the systems. Typically, such irregularities disappear due to the coupling components, but in this case, their presence has a significant influence on the whole pattern. Indeed, when the almost synchronized bobs reach the unstable equilibrium of the system, i.e. point \(\varphi _1^1 = \varphi _2^1 = \varphi _1^2 =\varphi _2^2 = -\pi \) (marked by the orange arrow), the first bob crosses the point, while the second turns back (blue and cyan trajectories, respectively). At this point, the synchronization is broken and the double pendula begin to move independently.

## Double Cross (2020) 01.mp4

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