In my research studying complicated graph-based network, I confront the following highly nonlinear and coupled ODE and I really have no idea how to take a further step:
Just take two nodes as an example:
Variables: $a_1,a_2,\theta_1,\theta_2$, all of which represent amplitude and phase of node one and two. (two nodes graph with the coupling $\beta$, ($0\leq\beta\leq 1$)
$$\dot{a}_1=a_1(1+\beta)-a_1^3-\beta a_2\cos(\theta_2-\theta_1)$$ $$\dot{a}_2=a_2(1+\beta)-a_2^3-\beta a_1\cos(\theta_1-\theta_2)$$ $$\dot{\theta}_1=\beta \frac{a_2}{a_1}\sin(\theta_1-\theta_2)$$ $$\dot{\theta}_2=\beta \frac{a_1}{a_2}\sin(\theta_2-\theta_1)$$
How to solve $a_1,a_2,\theta_1,\theta_2$?
If it is difficult to solve it, then is there any other way to study such ODE?
They look symmetric, is there any way to analyze it?
Once non-linear oscillators, coupling, and networks are involved, you can forget about solving this analytically. So, your first step would be to numerically solve your differential equations for some initial conditions and control parameter settings to get an idea how the dynamics behaves. What you do after that, depends on what kind of dynamics you observe and what you are interested in. For example, you can:
Treat your system like a single dynamical system and compute Lyapunov exponents, perform bifurcation analyses and stuff.
Investigate the global synchronisation behaviour using the master stability approach or transversal Lyapunov exponents.
Treat the system like an experiment and analyse macroscopic properties of the dynamics statistically and phenomenologically.
Relate your model to its motivation and investigate whether it reproduces real-world observations.