Maybe this is nonsense and wrong but if we calculate the spectrum of the Lyapunov exponents (LEs), we can assign each LE to each node, right?
If we have heterogeneous oscillators, we can have some information about the LE of each node.
Another problem is that when we use methods for calculation of the LEs like method introduced by Wolf and Sandri:
- Determining Lyapunov exponents from a time series, A Wolf, JB Swift, HL Swinney, JA Vastano - Physica D: Nonlinear Phenomena, 1985
- Numerical calculation of Lyapunov exponents, M Sandri - Mathematica Journal, 1996
the LEs are automatically sorted from largest to smallest, so we don't have information about each node.
Is there any equivalent method to resolve this issue?
Edit:
To give an example: I consider a system of coupled Kuramoto oscillators: $$ \frac{d\theta_i}{dt} = \omega_i + K / N \sum_{j=1}^N a_{ij} \sin(\theta_j(t) - \theta_i(t)) $$
where $\theta_i(t)$, phase of each oscillator at each time, $\omega_i$ are intrinsic angular frequency of oscillators, K is overall coupling and N is the number of oscillators, $a_{ij}$ are elements of adjacency matrix could be a weighted matrix (symmetric or not) and indicate the strength of connections between each pair of oscillators.
When I use Wolf method, I get something like this:
that each line is time evolution of each LEs, and red and blue colors indicate the positive or negative sign of each LE, respectively.
The difference between nodes is their intrinsic frequencies and also the connections going out of them. Some oscillators only allowed to have negative outward connections and the others only positive outward edges.