I am looking for literature on specific type of degeneracy for odes.
Consider the phase space $M= \{ (x_1,x_2)\in \mathbb{R}^2 \}$ and the ode of the general form: \begin{align} \alpha \frac{d x_1}{dt} &= f_1(x_1,x_2) \\ \frac{d x_2}{dt} &= f_2(x_1,x_2) \\ \end{align} with $f_1,f_2$ Lipschitz and $\alpha \in \mathbb{R}$ a parameter. I am interested in the $\alpha$-dependence of the solutions $(x_1,x_2)$. Specifically, in what happens to the solutions in the limit $\alpha \rightarrow 0$ where the ode becomes 1-dimensional (For example, does this happen uniformly?).
I am sure that these odes have been studied before. However, I am not able to find what they are called.
Any help is welcome.
For small $\alpha$, the system is called singularly perturbed. The solutions to singularly perturbed dynamical systems are studied using multiple time scale analysis and/or geometric singular perturbation theory. For more information on this subject, and especially the relation between the system when $\alpha$ is small but nonzero, and when $\alpha$ is identically zero, I highly recommend
C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015.