Energy exchange in globally coupled mechanical phase oscillators

We study the stationary dynamics of energy exchange in an ensemble of phase oscillators, coupled through a mean-field mechanical interaction and added with friction and an external periodic excitation. The degree of entrainment between different parts of the ensemble and the external forcing determines three dynamical regimes, each of them characterized by specific rates of energy exchange. Using suitable approximations, we are able to obtain analytical expressions for those rates, which are in satisfactory agreement with results from numerical integration of the equations of motion. In some of the dynamical regimes, the rates of energy exchange show nontrivial dependence on the friction coefficients—in particular, nonmonotonic behavior and sign switching. This suggests that, even in this kind of stylized model, power transfer between different parts of the ensemble and to the environment can be manipulated by a convenient choice of the individual oscillator parameters.

Figure 1

Schematic representation of the long-time dependence of phases for the three regimes of oscillation entrainment analyzed in the text. The dashed line stands for the phase of the external force, which moves at constant frequency in all cases. The full bold line corresponds to oscillator 1. The thin lines in different shades stand for the phases of five selected oscillators in the $\mathrm{\Omega }$ set.

Figure 2

Rates of energy exchange—$w_n^{\mathrm{\Omega }}, w_n^{\mathrm{\Gamma }}$, and $w_n^F$—in regime 1a, as functions of the damping coefficients of individual oscillators in the $\mathrm{\Omega }$ set. Lines correspond to the analytical results of Eqs. (36), and symbols show numerical results for $N=100$, both for $\overline{K}=10$ and $F=100$. The damping coefficients were drawn from a normal distribution with mean $\langle \gamma \rangle =0.2$ and standard deviation ${\sigma }_{\gamma }=0.05$, yielding ${\gamma }_1=0.22$.

Figure 3

Normalized rates of energy exchange—$u_n^{\mathrm{\Omega }}, u_n^{\mathrm{\Gamma }}$, and $u_n^F$—in regime 1b, as functions of the damping coefficients of individual oscillators in the $\mathrm{\Omega }$ set. Lines and symbols, respectively, correspond to analytical, Eqs. (38), and numerical results for $N=100, F=100$. Upper panel, case A: $\overline{K}=0.8, \langle \gamma \rangle =0.2$. Middle panel, case B: $K=3, \langle \gamma \rangle =2$. Lower panel, case C: $K=1.2, \langle \gamma \rangle =0.2$. In cases A and C, the numerical values of ${\gamma }_n$ were the same as in Fig. 2. In case B, the standard deviation was ${\sigma }_{\gamma }=0.5$, and ${\gamma }_1=1.6$.

Figure 4

As in Fig. 3 but for regime 2, with $F=0.5$ and ${\gamma }_1=10$ in the three cases.