Robust Weak Chimeras in Oscillator Networks with Delayed Linear and Quadratic Interactions

We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two populations of (at least) two elements using a general method based on a delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model-based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge chimera dynamics in phase models and real-world oscillator networks.

Figure 1

Weak chimeras exist for two coupled populations (2) of $N=2$ oscillators. Panel (a) shows the phase plane of the reduced system (3) for $ϵ=0.1$ (shading of arrows indicates their norm): Stable phase synchronized solutions in $SS$ are shown in red, the stable weak chimera periodic orbit in blue. Unstable periodic orbits (gray) bound both $\mathcal{B}(SS)$ (shaded area) and the basin of attraction of the weak chimera. In panel (b), the shading indicates the fraction of initial conditions $\mathbf{(}{\psi }_1(0),{\psi }_2(0)\mathbf{)}$ that lie in $\mathcal{B}(SS)$. Separation of average angular frequencies (${\mathrm{\Omega }}_{1,k}$ in blue and ${\mathrm{\Omega }}_{2,k}$ in red) characterizes the weak chimeras for (2) with fixed initial condition $\mathbf{(}{\theta }_{1,1}(0),{\theta }_{1,2}(0),{\theta }_{2,1}(0),{\theta }_{2,2}(0)\mathbf{)}=(0,\pi ,\pi /2,\pi /2+0.1)$. Note that, while the stable weak chimera exists up to $ϵ\approx 0.64$, this initial condition leaves its basin of attraction at $ϵ\approx 0.5$.

Figure 2

Network interaction yields weak chimeras in Brusselator models for $A=1$ and $B=2.3$. Panel (a) shows the waveform $x$, phase response curve $Z$, and the effective interaction function $g_{\mathrm{Br}}$ (dotted) on top of the target (1) with $\alpha =0$ (solid) for second-order feedback parameters $({\tau }_1,{\tau }_2)=2\pi {\omega }_{\mathrm{Br}}^{-1}(0.8577,0.3156)$ and $(k_1,k_2)=(0.6601,-2.1692)$. Panel (b) depicts waveforms $x_{\sigma ,k}$ and phase differences $⟨{\varphi }_{k,\sigma }-{\varphi }_{2,2}⟩$ of a weak chimera for this feedback, $ϵ=0.1$, and $K=0.03$: The frequencies of populations 1 (blue) and 2 (red) are distinct. These weak chimeras exist for a range of $ϵ$ as shown in panel (c), here $K=0.05$. The initial condition was the point with $\mathbf{(}{\theta }_{1,1}(0),{\theta }_{1,2}(0),{\theta }_{2,1}(0),{\theta }_{2,2}(0)\mathbf{)}=(0,\pi ,\pi /2,\pi /2+0.1)$ for the uncoupled system.

Figure 3

Experimental weak chimera; $V_0=1160\mathrm{mV}$, $R_{\mathrm{ind}}\simeq 1\mathrm{k}\mathrm{\Omega }$. (a) Experimentally determined (points) and desired (line) interaction function for $\tau =0$, $r=-0.4$; $K=0.35$. (b) Phase difference time series of population 1 (${\varphi }_{1,2}-{\varphi }_{1,1}$) and population 2 (${\varphi }_{2,2}-{\varphi }_{2,1}$) and between the populations (${\varphi }_{2,2}-{\varphi }_{1,1}$) of a weak chimera; $K=0.52$, $ϵ=0.1$. (c) Differences of frequencies (averaged over populations) between populations without coupling, $K=0$ (circles) and at various $ϵ$ values, $K=0.52$ (squares). (d) The frequencies of the populations at various $ϵ$ values ($\mathrm{squares}=\text{population}$ 1, $\mathrm{circles}=\text{population}$ 2) at $K=0.52$.