Self-Organized Quasiperiodicity in Oscillator Ensembles with Global Nonlinear Coupling

We describe a transition from fully synchronous periodic oscillations to partially synchronous quasiperiodic dynamics in ensembles of identical oscillators with all-to-all coupling that nonlinearly depends on the generalized order parameters. We present an analytically solvable model that predicts a regime where the mean field does not entrain individual oscillators, but has a frequency incommensurate to theirs. The self-organized onset of quasiperiodicity is illustrated with Landau-Stuart oscillators and a Josephson junction array with a nonlinear coupling.

Figure 1

Simulation of Eqs. (9, 10). Left panels: Order parameter $K$ (circles), frequencies of oscillator ${\omega }_{\mathrm{osc}}$ (squares) and of the mean field $\Omega$ (diamonds) vs coupling strength $\epsilon$. Corresponding theoretical curves are shown by solid lines and nearly coincide with numerics. Normalized minimal distance $\Delta$ (triangles) demonstrates the absence of clusters in the quasiperiodic regime. Right panel shows a snapshot of the ensemble (circles), which can be considered as a sampled distribution (7); star is the mean field.

Figure 2

Transition to self-organized quasiperiodicity in an array of nonlinearly coupled Josephson junctions (12, 13). The order parameter $K=|Z_1|=|⟨e^{i\theta }⟩|$ [(a), solid line] and the frequencies of the mean field [(b), dashed line] and of individual oscillators [(b) solid line] are shown as functions of dimensionless coupling strength $\epsilon =N\hslash (2e{I}_c{L}_0{)}^{-1}$. Other dimensionless parameters are $I/I_c=5$, $\hslash /[\sqrt{L_{0}C}(2er{I}_c)]=4.8$, $L_1/(L_0{I}_c^2)=-0.5$, $2erR{I}_c/(L_0\hslash )=0.05$. Insets demonstrate scaling ($1-K$), $\Omega -⟨\dot{\theta }⟩\propto \epsilon -{\epsilon }_q$. Stability of fully synchronous state is quantified by multiplier $\mu$ (see text), shown by dashed line in (a); transition to quasiperiodicity appears exactly when $\mu$ becomes large than one.