Heteroclinic switching between chimeras

Functional oscillator networks, such as neuronal networks in the brain, exhibit switching between metastable states involving many oscillators. We give exact results how such global dynamics can arise in paradigmatic phase oscillator networks: Higher-order network interactions give rise to metastable chimeras—localized frequency synchrony patterns—which are joined by heteroclinic connections. Moreover, we illuminate the mechanisms that underly the switching dynamics in these experimentally accessible networks.

Figure 1

Heteroclinic networks appear in networks of $M=3$ populations of $N=2$ oscillators (5) with coupling function (2) and $\alpha =\frac{\pi }{2},r=-0.1,K=0.1$. Panel (a) shows the heteroclinic cycle between saddle weak chimeras (the solid lines); stability is indicated by arrows. Panel (b) shows switching of localized frequency synchrony close to the heteroclinic cycle for $\delta =0$ in the presence of noise $\eta ={10}^{-7}$: The oscillators' phases (shading; black if ${\theta }_{\sigma ,k}=\pi$, white if ${\theta }_{\sigma ,k}=0$) and frequencies (colors; shading indicates a population's frequency range $[{min}_k{\dot{\theta }}_{\sigma ,k},{max}_k{\dot{\theta }}_{\sigma ,k}]$, a line its average ${\langle {\dot{\theta }}_{\sigma ,k}\rangle }_k$) are plotted over time. The frequencies of different populations synchronize and desynchronize sequentially. Panel (c) shows irregular switching dynamics without noise $\eta =0$ when the symmetries are broken, $\delta =0.01$.

Figure 2

Switching between localized frequency synchrony is observed in networks of $M=3$ populations of $N=11$ oscillators with dynamics (5) and vector field (11). As in Fig. 1 the evolution of phases and order parameters the oscillators populations synchronize in frequency sequentially. Coupling for $K=0.2$ is given by (6) with $a=22,r=-0.001,\alpha =\frac{\pi }{2}$, symmetry breaking $\delta =0.001$ and no noise $\eta =0$.

Figure 3

The network of $M=4$ populations of $N=2$ oscillators with dynamics (13) shows noise induced random switching from SDSS to either SSDS or SSSD. This relates to a Kirk-Silber type network sketched in panel (a). Panel (b) depicts evolution of phases and frequencies (populations 3 and 4 are highlighted in color) for coupling (2) with $r=-0.1,\alpha =\frac{\pi }{2},K=0.35$, and $\eta ={10}^{-4}$.