==synchrotools==

synchrotools is a python package providing a collection of indices that characterize

• synchronization
• chimera states
• metastable states

Data are handled with numpy ndarrays and dictionaries

• Kuramoto order Parameter

• Global order parameter: \begin{equation} \rho = \frac{1}{N}\sum_{j=1}^N e^{\mathbf{i}\theta_j(t)} \end{equation}

• Local order parameter: \begin{equation} \rho_i = \frac{1}{2\delta}\sum_{j=i-\delta}^{i+\delta} e^{\mathbf{i}\theta_j(t)} \end{equation}

• Mean correlation

• Mean phase velocity \begin{equation} \Omega_i = \frac{2 \pi K_i}{\Delta T},, \end{equation} where (K_i) is the number of periods of the $i$-th oscillator during a time interval (\Delta T).

• Metastabilite index (\lambda):

\begin{equation} \lambda = {\langle \sigma_{\textbf{met}} \rangle}{C_m}\end{equation} where \begin{equation} \sigma{\textbf{met}}(m)=\frac{1}{T-1}\sum_{t=1}^T(\rho_m(t)-{\langle \rho_m \rangle}_T)^2\end{equation}

• Chimera-like index ( \chi ) \begin{equation} \chi={\langle \sigma_{\textbf{chi}} \rangle}T, \end{equation} where \begin{equation} \sigma{\textbf{chi}}(t)=\frac{1}{M-1}\sum_{m=1}^M(\rho_m(t)-{\langle \rho(t) \rangle}_M)^2 \end{equation}

• Local curvature

• Local curvature: \begin{equation} \hat{D}\theta_i(t) := \sum_{j=i-\delta}^{j=i+\delta} !\left[\theta_j(t) - \theta_i(t)\right], \end{equation}

• Spatial coherence:

The norm of the operator (\hat{D}) in the coherent clusters is zero (or sufficiently small) ( |\hat{D}{\theta_{i_\text{coh}}(t)}|\approx 0 ), while in the incoherent clusters ( |\hat{D}{\theta_{i_\text{incoh}}(t)}| ) is finite and has pronounced fluctuations. The maximum value ( D_m ) of ( |\hat{D}\theta_i(t)| ) corresponds to the local curvature of nodes whose neighbors.