Optimization of noise-induced synchronization of oscillator networks

We investigate common-noise-induced synchronization between two identical networks of coupled phase oscillators exhibiting fully locked collective oscillations. Using the collective phase description method for fully locked oscillators, we demonstrate that two noninteracting networks of coupled phase oscillators can exhibit in-phase synchronization between the networks when driven by weak common noise. We derive the Lyapunov exponent characterizing the relaxation time for synchronization and develop a method of obtaining the optimal input pattern of common noise to achieve fast synchronization. We illustrate the theory using three representative networks with heterogeneous, global, and local coupling. The theoretical results are validated by direct numerical simulations.

Figure 1

Two-oscillator system. Dependence of the left zero eigenvector and optimal input pattern, i.e., $v_1, v_2, a_1$, and $a_2$, on the parameter $\eta$ under the phase shift $\alpha =\pi /4$. The normalization conditions are $v_1+v_2=1$ and $a_1^2+a_2^2=1$.

Figure 2

Heterogeneously coupled system. The coupling network is generated by the Barabási-Albert model with $N=100$ and $m=m_0=2$, the oscillators are identical, i.e., ${\omega }_j=\omega$, and the phase shift is $\alpha =\pi /8$. (a) Network structure associated with the symmetric adjacency matrix $A_{jk}=A_{kj}$. (b) Degree $d_j={\sum }_{l\ne j}{A}_{jl}={\sum }_{l\ne j}{A}_{lj}$. (c) Effective adjacency matrix, i.e., $-{\mathrm{\Gamma }}_{jk}^{\prime }=A_{jk}cos({\psi }_j-{\psi }_k+\alpha )$, in the fully locked state. (d) Effective in-degree $d_j^{\mathrm{in}}={\sum }_{l\ne j}{A}_{jl}cos({\psi }_j-{\psi }_l+\alpha )$ and effective out-degree $d_j^{\mathrm{out}}={\sum }_{l\ne j}{A}_{lj}cos({\psi }_l-{\psi }_j+\alpha )$.

Figure 3

Heterogeneously coupled system. The coupling network is generated by the Barabási-Albert model with $N=100$ and $m=m_0=2$, the oscillators are identical, i.e., ${\omega }_j=\omega$, and the phase shift is $\alpha =\pi /8$. (a) Degree $d_j$. (b) Relative phase ${\psi }_j$. (c) Left zero eigenvector $v_j$. (d) Input pattern $a_j$. The optimal input pattern ${\mathit{a}}_{\mathrm{opt}}$ and approximate input pattern ${\mathit{a}}_{\mathrm{app}}$ are shown. Note that the uniform input pattern ${\mathit{a}}_{\mathrm{uni}}$ is a vector whose components are all $1/\sqrt{N}=0.1$. Note also that when the common noise is applied only to the hub, the input pattern is ${\mathit{a}}_{\mathrm{hub}}={\mathbf{e}}_1$.

Figure 4

Globally coupled system. The parameters are $N=100, {\omega }_{\delta }=0.5$, and $\alpha =\pi /4$. (a) Natural frequency ${\omega }_j$ under $\omega =0$. (b) Relative phase ${\psi }_j$. (c) Left zero eigenvector $v_j$. (d) Input pattern $a_j$. The optimal input pattern ${\mathit{a}}_{\mathrm{opt}}$ and approximate input pattern ${\mathit{a}}_{\mathrm{app}}$ are shown. Note that the uniform input pattern ${\mathit{a}}_{\mathrm{uni}}$ is a vector whose components are all $1/\sqrt{N}=0.1$.

Figure 5

Locally coupled system. The parameters are $N=200, {\omega }_p=0.25$, and $\alpha =\pi /12$. (a) Natural frequency ${\omega }_j$ under $\omega =0$. (b) Relative phase ${\psi }_j$ scaled by $\pi$. (c) Left zero eigenvector $v_j$. (d) Input pattern $a_j$. The optimal input pattern ${\mathit{a}}_{\mathrm{opt}}$ and approximate input pattern ${\mathit{a}}_{\mathrm{app}}$ are shown. Note that the uniform input pattern ${\mathit{a}}_{\mathrm{uni}}$ is a vector whose components are all $1/\sqrt{N}\simeq 0.07$.

Figure 6

Locally coupled system. Snapshots of $N{Z}_j\left(\mathrm{\Theta }\right)=-N{v}_{j}sin(\mathrm{\Theta }+{\psi }_j)$ for $\mathrm{\Theta }/\pi =0.0$, 0.5, 1.0, and 1.5. The parameters are $N=200, {\omega }_p=0.25$, and $\alpha =\pi /12$.

Figure 7

Comparison of direct numerical simulations (symbols) and theoretical values (lines) of the Lyapunov exponents for several input patterns (i.e., ${\mathit{a}}_{\mathrm{opt}}, {\mathit{a}}_{\mathrm{app}}, {\mathit{a}}_{\mathrm{uni}}$, and ${\mathit{a}}_{\mathrm{hub}}$). The initial collective phase difference is $|{\mathrm{\Theta }}^{\left(1\right)}(t=0)-{\mathrm{\Theta }}^{\left(2\right)}(t=0)|={10}^{-1}$. The base frequency $\omega$ is chosen such that the collective frequency becomes unity, $\mathrm{\Omega }=1$. The noise intensity is ${\epsilon }^2=0.0025$. The results of direct numerical simulations are averaged over 500 samples. (a) Heterogeneously coupled system. (b) Globally coupled system. (c) Locally coupled system.

Figure 8

Heterogeneously coupled system with a negative phase shift. The coupling network is generated by the Barabási-Albert model with $N=100$ and $m=m_0=2$, the oscillators are identical, i.e., ${\omega }_j=\omega$, and the phase shift is $\alpha =-\pi /8$. (a) Degree $d_j$. (b) Relative phase ${\psi }_j$. (c) Left zero eigenvector $v_j$. (d) Input pattern $a_j$. The optimal input pattern ${\mathit{a}}_{\mathrm{opt}}$ and approximate input pattern ${\mathit{a}}_{\mathrm{app}}$ are shown. Note that the uniform input pattern ${\mathit{a}}_{\mathrm{uni}}$ is a vector whose components are all $1/\sqrt{N}=0.1$. Note also that when the common noise is applied only to the hub, the input pattern is ${\mathit{a}}_{\mathrm{hub}}={\mathbf{e}}_1$.

Figure 9

Globally coupled system with a negative phase shift. The parameters are $N=100, {\omega }_{\delta }=0.5$, and $\alpha =-\pi /4$. (a) Natural frequency ${\omega }_j$ under $\omega =0$. (b) Relative phase ${\psi }_j$. (c) Left zero eigenvector $v_j$. (d) Input pattern $a_j$. The optimal input pattern ${\mathit{a}}_{\mathrm{opt}}$ and approximate input pattern ${\mathit{a}}_{\mathrm{app}}$ are shown. Note that the uniform input pattern ${\mathit{a}}_{\mathrm{uni}}$ is a vector whose components are all $1/\sqrt{N}=0.1$.