I'm having trouble understanding the concept of degeneracy in linear and nonlinear systems. I have read chapters where system stability is discussed in my linear algebra/controls engineering textbooks, but the definition of degeneracy is not discussed in any of them. For instance, if you have the nonlinear system
$ \begin{cases} \dot{x}_1 = f_1(x_1, x_2)\\ \dot{x}_2 = f_2(x_1, x_2)\\ \end{cases} $
and the system is linearized to resemble $\dot{x} \approx Ax + Bu$, the eigenvalues of $A$ can be computed to determine stability. So, if $\lambda_1 = 0$ and $\lambda_2 < 0$, the system is a stable degenerate node; similarly if $\lambda_1 = 0$ and $\lambda_2 > 0$, the system is an unstable degenerate node. In this context, what does degeneracy mean? Thank you in advance!
Degenerate in this instance means that one of the eigenvalues is zero.
If a linear system is degenerate, it means the steady state is not isolated. There is a one-parameter family of steady states corresponding to the eigenvector with zero eigenvalue.
If the linearization of a nonlinear system is degenerate, it means that the linearization does not contain all the information on how solutions behave near the steady state; a zero eigenvalue could correspond to (sub-exponential) growth or decay, or the existence of one-parameter family of steady states.