Diffusion-Induced Inhomogeneity in Globally Coupled Oscillators: Swing-By Mechanism

It is shown that for random initial conditions, a large population of identical and sufficiently nonisochronous Stuart-Landau oscillators coupled globally and diffusively exhibits inhomogeneity in a resonant way as the diffusive coupling is intensified, where the diffusive coupling constant is real. A category of inhomogeneous (nonsynchronized) solutions is analytically shown to exist, which is either periodic or quasiperiodic.

Figure 1

The behavior of $\sigma$ for $N=4000$ averaged over ten realizations of the random initial condition explained in text. The numbers attached to the data are the values of $c_2$. The lines connecting the symbols are to guide the eye. The inset shows $K\chi$’s graphs in the range $0\le K\le 1$, $0\le K\chi \le 0.6$ for the same values of $c_2$, where its maximum values as well as its maximum points indicated by the vertical dotted lines in the main panel monotonically decrease for increasing $|c_2|$.

Figure 2

Examples of desynchronized solutions for $c_2=-3$, $N=1000$. There are two synchronized sets of oscillators; the bigger (size $N-M$) and the smaller (size $M$) are drawn by real and broken curves, respectively. (a) $K=0.51$ (quasiperiodic, $M=4$). (b) $K=0.94$ (periodic, $M=9$).

Figure 3

Relaxation from a random initial condition for $c_2=-3$, $K=0.65$, $N=1000$. (a) $r_j\equiv |z_j|$, where the horizontal broken line shows the radius of the incoherent circle. (b) ${\theta }_j\equiv \mathrm{Arg}(z_j)$. (c) Order parameters $R$ and $Q$. The inset shows $R$ vs $Q$ over the ranges $0<Q$, $R<1$, where the dotted line shows $R=\chi Q$.

Figure 4

The cluster structure of numerically found solutions for a random initial condition and $c_2=-3$, $N=1000$. Each segment of a vertical bar shows the size ratio of a cluster smaller than the main cluster, which is not shown here. These segments are placed in the order of their sizes, from top to bottom. Almost black regions correspond to the existence of a large number of small clusters. A cluster here is defined to be a collection of oscillators less than ${10}^{-5}$ apart from one oscillator at $t=10000$. The behavior of $\sigma$ is also shown.

Figure 5

The phase diagram based on the approximation developed in text, where the rightmost, middle, and leftmost curves show $K=K_{\mathrm{SN}},K_{\mathrm{H}},K_{\mathrm{SC}}$, respectively. Note that the diagram does not depend on the sign of $c_2$. The symbols show those points which lead to an inhomogeneous solution in simulation of Eq. (1) for a random initial condition, where the nature of the solution is periodic (solid circle) or nonperiodic (cross).