Light-powered Self-excited Coupled Oscillators in Huygens ’ Synchrony


 Self-excited motions have the advantages of actively collecting energy from the environment, autonomy, making equipment portable and so on, and a great number of self-excited motion modes have recently been developed which greatly expand the application in active machines. However, there are few studies on the synchronization and group behaviors of self-excited coupled oscillators, which are common in nature. Based on light-powered self-excited oscillator composed of liquid crystal elastomer (LCE) bars, the synchronization of two self-excited coupled oscillators is theoretically studied. Numerical calculations show that self-excited oscillations of the system have two synchronization modes: in-phase mode and anti-phase mode. The time histories of various quantities are calculated to elucidate the mechanism of self-excited oscillation and synchronization. Furthermore, the effects of initial conditions and interaction on the two synchronization modes of the self-excited oscillation are investigated extensively. For strong interactions, the system always develops into in-phase synchronization mode, while for weak interaction, the system will evolve into anti-phase synchronization mode. Meanwhile, the initial condition generally does not affect the synchronization mode and its amplitude. This work will deepen people's understanding of synchronization behaviors of self-excited coupled oscillators, and provide promising applications in energy harvesting, signal monitoring, soft robotics and medical equipment.

period of the self-excited oscillations are investigated. Finally, the conclusion is given 89 in Sec. 6.   132 where 0 C is the contraction coefficient. The number fraction ) (t  will be given in  The moment exerted by the torsion spring on the two bars is assumed to be 146 linear with the angle difference, 148 where  is the spring coefficient of the torsion spring. neglected. Therefore, the number fraction of cis isomers depends on the thermal 156 excitation from trans to cis, the thermally driven relaxation from cis to trans and the where 0 T is the thermal relaxation time of cis state to trans state, 0 is the light intensity, and 0  is the light absorption constant. By solving Eq. (10), the number 162 fraction of cis-isomers can be expressed as: In the dark zone, namely 0 0  I , 0  can be set as the maximum value of (12), namely, 1 , and Eq. (11) can be simplified as: By defining the following dimensionless quantities: , in the light zone,

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The initial conditions are as follows： (21) 200 By defining the following vectors , and , Eqs. (20) and (21) can be expressed as follows: Following the classical fourth-order Runge-Kutta method, Eq. (22) can be written as,

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where H is the time step, and i k ( i =1 to 4) are listed as below, oscillators: in-phase mode and anti-phase mode. In the computation, we fix . In the following, we will 221 discuss the two synchronization modes in turn. oscillators. The parameters are .
. In Figs. 4a and   and 2  evolve from the initial disorder into a final attraction domain. Fig. 4e further plots the domains of attraction of 1  and 2  for different initial conditions.

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The calculation shows that the two domains of attraction are the same. The results

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show that the initial condition has no effect on the synchronous mode.   constants. In the computation, we set . In  Furthermore, Fig. 5f describes the limit cycles of 1  and 1   for different spring 304 constants. Similarly, the two limit cycles are also the same. This implies that the 305 spring constant have no effect on the amplitude and frequency of the LCE oscillators.
. This is because that for crit    , the LCE oscillators are in in-phase 308 mode, and both the angle difference between two bars and the moment of spring are 309 zero. Therefore, spring constant has no effect on the its amplitude and frequency. In 310 in-phase mode, the system is equivalent to the single oscillator, as shown in Fig. 6a.
. In Figs. 8a .  we set , the phase difference between the two bars after a period of time is a constant value that is generally not  180 or  0 , as shown in Fig. 9a. Fig. 9d