Experimental multistable states for small network of coupled pendula

Chimera states are dynamical patterns emerging in populations of coupled identical oscillators where different groups of oscillators exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Although these states are typically observed in the large ensembles of oscillators, recently it has been shown that so-called weak chimera states may occur in the systems with small numbers of oscillators. Here, we show that similar multistable states demonstrating partial frequency synchronization, can be observed in simple experiments with identical mechanical oscillators, namely pendula. The mathematical model of our experiment shows that the observed multistable states are controlled by elementary dynamical equations, derived from Newton’s laws that are ubiquitous in many physical and engineering systems. Our finding suggests that multistable chimera-like states are observable in small networks relevant to various real-world systems.


Results
In the absence of coupling (when one removes green planar springs and thus, coupling parameter α = 0 in Equation (1) in Methods) it is possible to identify excitation parameters (A and ω ) for which each double pendulum exhibits multistability. In Fig. 2 we present regions of existence of various N:M , where N is the number of rotation/oscillation of lower pendulum II [1][2][3][4] and M is the number of periods of excitations, eg., 1:1 means that pendula II 1-4 oscillate or rotate with the frequency of the excitation ω, 2:1 (pendula II 1-4 oscillate or rotate with the frequency of the excitation ½ ω), etc. One can identify six main regions, indicated from 1 to 6 in Fig. 2, in which the excited double pendulum is multistable. In region 1 three solutions exist: 1:1 rotations (above the green line), 1:4 oscillations (between the dashed red lines) and 1:2 rotations (between solid black lines). Region 2 is characterized by the co-existence of four solutions: 1:1 rotations (above the green line), 1:4 oscillations (between the dashed red lines), 1:2 rotations (between solid black lines) and 3:6 rotations (between solid orange lines). Four solutions are stable also in region 3:1:1 rotations (above the green line), 1:6 oscillations (between the dashed black lines), 1:2 rotations (between solid black lines), 1:3 rotations (between solid yellow lines). Three solutions: 1:1 rotations (above the green line), 1:6 oscillation (between dashed black lines), 1:2 rotation (between solid black lines) can be observed in region 4. Region 5 is another example of the co-existence of three solutions: 1:1 rotations (above the green line), 1:2 rotations (between solid black lines), 1:3 rotation (between solid yellow lines). Finally in region 6 we observe four solutions: 1:1 rotations (above green line), 1:4 oscillations (between the dashed black lines), 3:6 rotations (between solid orange lines).
In regions 1-6 each of four uncoupled double pendula can exhibit M (equal to 3 or 4) various independent dynamical responses, i.e., 1:1, 2:1 or 3:1 rotational and oscillatory solutions. The set of 4 pendula is characterized by M 4 configurations. One can see that the number of configurations grows exponentially with the number of pendula (i.e., in the case of n pendula we have M n configurations) so there is spatial chaos 31 in an uncoupled system. For sufficiently small coupling one can observe multistable chimera-like states which persist over the wide range of system parameters and can be captured experimentally. These states coexist with various cases of complete, phase and cluster synchronous states.
Experimentally observed multistable chimera-like states are illustrated in Fig. 3(a-f). Upper images present general view of the pendula's configurations while lower plots show time series of the lower pendula bobs. The  figures present a kind of a stroboscope type images of the pendula motion in different cases. All experiments have been recorded using Vision Research Phantom v711 high speed camera. Typical recording speed was 1000 frames per second (fps) and for the purpose of a still photograph visualization a set of 5 of them every fifth frame: 5 × 0.001 = 0.005 seconds have been chosen. Then, the images were combined to a single image presenting all chosen images overlaid with the assumed transparency level. The wider area covered by the set of frozen images of each pendulum, the faster speed of its rotation or oscillation and vice verse. In Fig. 3(a-d) we show multistable states in which all the pendula rotate (A = 0.01 [m], ω = 18π [rad/s]-region 5 of Fig. 2). In Fig. 3(a) pendula 1 and 2 rotate with frequency ω 1 3 and pendula 3 and 4 with frequency ω.  and pendulum 2 with frequency ω 1 3 , (c) pendulum 1 rotates with frequency ω 1 3 , pendula 2 and 3 with frequency ω and pendulum 4 with frequency ω 1 2 , (d) pendula 1, 2 and 4 rotate with frequency ω 1 3 and pendulum 3 with frequency ω, (e) pendula 1, 3 and 4 rotate with frequency ω 1 2 , pendulum 2 oscillates with frequency ω 1 4 , (f) pendula 1 and 4 rotate with frequency ω , pendulum 2 rotates with frequency ω 1 2 and pendulum 3 oscillates with the frequency ω 1 4 . other (see movie W1). The case in which pendula 1, 3 and 4 rotate with frequency ω 1 2 and pendulum 2 with frequency ω 1 3 is shown in Fig. 3(b). Pendula 1 and 4 are synchronized in phase and pendulum 3 is in antiphase to pendula 1 and 4 (see movie W2). Configuration of Fig. 3(c) presents the case when pendulum 1 rotates with a frequency ω 1 3 , pendula 2 and 3 with frequency ω and pendulum 4 with frequency ω 1 2 . Pendula 2 and 3 are synchronized (see movie W3). Figure 3(d) shows the configuration in which pendula 1, 2 and 4 rotate with frequency ω 1 3 and pendulum 3 with frequency ω. Pendula 1 and 2 are synchronized in phase (see movie W4). In Fig. 3(e,f) we observe multistable states in which the pendula show both rotational and oscillatory behavior (A = 0.005 [m], ω = 10π [rad/s]-region 1 of Fig. 2). Figure 3(e) shows the chimera-like state in which pendula 1, 3 and 4 rotate with frequency ω 1 2 while pendulum 2 oscillates with frequency ω 1 4 . Pendula 1 and 4 are synchronized in phase (see movie W5). The chimera-like state shown in Fig. 3(f) is characterized by 3 rotating and one oscillating pendula. Pendula 1 and 4 rotate with the frequency ω and are synchronized in phase. Pendula 2 and 3 respectively rotate with frequency ω 1 2 and oscillate with frequency ω 1 4 (see movie W6). The presented multistable states coexist with various synchronous states. Movies W7-W9 present the case of the complete synchronization of all pendula in rotational motion (W7), the case when all pendula oscillate with frequency ω and pendula 2, 3, 4 are synchronized in phase and pendulum 1 is in antiphase to them (W8) and the case when all pendula oscillate with the frequency ω and pendula 1, 3 and 2, 4 create two clusters of phase synchronized pendula respectively. These clusters are in antiphase to each other (W9).
In conclusion, we have constructed the simple experimental setup to explore the spatio-temporal dynamics of the small network of the locally coupled pendula. The nodes in the network are externally excited double pendula. Despite a small number of nodes, namely 4, we observe the formation of spatio-temporal patterns of multistable chimera-like states. This behavior is observed experimentally, confirmed in numerical simulations, persistent over a positive measure set of system parameters and seems to be characteristic for the small networks of coupled multistable general oscillators relevant to various real-world systems.

Methods
The dynamics of the system of coupled pendula shown in Fig. 1(a) is given by: Experimental observations. In our experiments, the rig with four coupled double pendula has been mounted on the shaker LDS V780 Low Force Shaker (basic data are as follows: sine force peak 5120[N], max random force (rms) 4230[N], max acceleration sine peak g n = 111 g [m/s 2 ], system velocity sine peak 1.9[m/s], displacement pk-pk g n = 25.4[mm], moving element mass 4.7[kg]). The shaker introduces practically kinematic periodic excitation ω A t cos , where A and ω are the amplitude and the frequency of the excitation, respectively. All experiments were recorded at motion videos taken by Vision Research Phantom v711 high speed camera. Typical recording speed used was 1000 frames per second (fps). Different random initial conditions have been given to each pendulum.