Low-dimensional dynamics of phase oscillators driven by Cauchy noise

Phase oscillator systems with global sine coupling are known to exhibit low-dimensional dynamics. In this paper, such characteristics are extended to phase oscillator systems driven by Cauchy noise. The low-dimensional dynamics solution agreed well with the numerical simulations of noise-driven phase oscillators in the present study. The low-dimensional dynamics of identical oscillators with Cauchy noise coincided with those of heterogeneous oscillators with Cauchy-distributed natural frequencies. This allows for the study of noise-driven identical oscillator systems through heterogeneous oscillators without noise and vice versa.

Figure 1

Numerical and theoretical results of $\langle \left|z\right|\rangle$ for the system of Eq. (5) with $K=1$. The numerical and theoretical results are indicated by the circles and the solid line, respectively.

Figure 2

Numerical results of $\langle \left|z\right|\rangle$ for the system of Eq. (27) with $K=1$. The theoretically derived boundary between the synchronized state and the desynchronized state is indicated by the dashed line.