Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on the overall coupling intensity and the stationary phase difference. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.

Figure 1

Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling function of coupled Stuart-Landau oscillators. The cases with identity coupling and optimal coupling are compared for $P=0.1$. Straight lines represent the slopes at the origin.

Figure 2

In-phase synchronization dynamics of coupled identical Stuart-Landau oscillators. The cases with identity coupling and optimal coupling are compared for $P=0.1$. Results obtained by direct numerical simulations (DNS) of the coupled Stuart-Landau oscillators and by numerically integrating reduced phase equations are shown.

Figure 3

In-phase synchronization dynamics of coupled Stuart-Landau oscillators for $P=0.1$ and $\epsilon =0.05$. Identity coupling (a) vs optimal coupling (b).

Figure 4

Limit-cycle solution ${\mathit{X}}_0\left(\theta \right)=(x_0\left(\theta \right),y_0\left(\theta \right){)}^{\mathrm{T}}$ (a) and phase sensitivity function $\mathit{Z}\left(\theta \right)=(Z_x\left(\theta \right),Z_y\left(\theta \right){)}^{\mathrm{T}}$ (b) of the Brusselator.

Figure 5

Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling function of coupled Brusselators. The results for the optimal and identity coupling matrices $K_{\mathrm{opt}}$ and $K_I$ are compared for $P=0.1$. Straight lines represent the slopes at the origin.

Figure 6

Evolution of phase difference $\varphi$ for identity and optimal coupling matrices, $K_I$ and $K_{\mathrm{opt}}$, starting from initial phase difference $\varphi =0.5$. Results obtained by direct numerical simulations (DNS) of coupled Brusselators and by numerically integrating the reduced phase equations are shown. The parameters are $P=0.1$ and $\epsilon =0.05$.

Figure 7

In-phase synchronization process of two identical Brusselators with identity (a) and optimal (b) coupling matrices, $K_I$ and $K_{\mathrm{opt}}$. In each figure, the time sequence of the difference in $x$ components between the two oscillators is plotted by a solid line, and the dashed line indicates an exponentially decaying curve with a decay rate $\epsilon {\mathrm{\Gamma }}_a^{\prime }\left(0\right)<0$ (actually $4exp\left[\epsilon {\mathrm{\Gamma }}_a^{\prime }\left(0\right)t\right]$). The parameters are $P=0.1$ and $\epsilon =0.05$.

Figure 8

Necessary conditions for the overall coupling intensity $P$ plotted as functions of the target phase difference ${\varphi }^{*}$ for frequency mismatch $\mathrm{\Delta }\omega =0.175$. Only nonshaded regions are realizable. The first condition (38) is violated in the red-shaded region, and the second condition (42) is violated in the blue-shaded region. Crosses represent the values of the given phase difference ${\varphi }^{*}$ used in the example.

Figure 9

Matrix elements of the optimal coupling matrix $K_{11},K_{12},K_{21},K_{22}$ and linear stability $-{\mathrm{\Gamma }}_a^{\prime }\left({\varphi }^{*}\right)$ plotted as functions of the phase difference ${\varphi }^{*}$ for $P=0.1$. Dotted vertical lines represent the boundaries of the regions in which both of the necessary conditions are satisfied.

Figure 10

Examples of the antisymmetric part of the phase-coupling function for given stable phase differences, ${\varphi }^{*}=2.0,1.5,1.0,0.5,-1.0,-1.5,-2.0,$ and $-2.5$, for frequency mismatch $\mathrm{\Delta }\omega =0.1754$. Crosses represent stable fixed points.

Figure 11

Convergence of phase differences to given values (${\varphi }^{*}=2.0,1.5,1.0,0.5,-1.0,-1.5,-2.0,$ and $-2.5$, indicated by horizontal dashed lines). Results of direct numerical simulations (DNS) are compared with the specified phase values for frequency mismatch $\mathrm{\Delta }\omega =0.175$ and $\epsilon =0.02$.

Figure 12

(a) Limit-cycle solution ${\mathit{X}}_0\left(\theta \right)=(x_0\left(\theta \right),y_0\left(\theta \right),z_0\left(\theta \right){)}^{\mathrm{T}}$ of the Lorenz model, where the $z$ variable is shifted by 300 for clarity. (b) Phase sensitivity function $\mathit{Z}\left(\theta \right)=(Z_x\left(\theta \right),Z_y\left(\theta \right),Z_z\left(\theta \right){)}^{\mathrm{T}}$ of the Lorenz model.

Figure 13

(a) Antisymmetric part ${\mathrm{\Gamma }}_a\left(\varphi \right)$ of the phase-coupling function of coupled Lorenz models. The straight line represents the slope at the origin. (b) Convergence of phase difference $\varphi \left(t\right)$ from $\varphi \left(0\right)=0.5$ to 0. Results of direct numerical simulations (DNS) and phase models are compared for $\epsilon =0.5$. In each figure, results for optimal and identity coupling matrices $K_{\mathrm{opt}}$ and $K_I$ are compared.

Figure 14

Collective synchronization of $N=400$ globally coupled Stuart-Landau oscillators with identical properties. Time sequences of the modulus of the Kuramoto order parameter $R$ for $K=K_I$ and $K=K_{\mathrm{opt}}$ are plotted ($P=0.1$ and $\epsilon =0.05$). The insets show typical snapshots of the oscillator distributions at the initial state and at the final state sufficiently after the convergence.