# To what extent is linear stability analysis of numerical methods relevant to nonlinear ODEs?

Some context.

Consider a numerical method $y_{n+1} = \Psi(y_n, h)$ for solving ODEs, where $h$ is the step size.

The following definition of linear stability is ubiquitous in the introductory numerical analysis literature on numerical ODEs (see Definition 8 in the these notes):

Suppose $y' = \lambda y$ for some $\lambda\in\mathbb C$. Then the numerical method $\Psi$ is linearly stable if $y_n\to 0$ as $n\to\infty$.

As a numerical methods novice, I wonder to what extent this definition is relevant to non-linear ODEs given that it is formulated in terms of a linear model problem. Later in the same notes, the following statement appears and seems to address this:

If a numerical method is stable in the above sense for a certain range of values of $\lambda$, then it is possible to show that it will be stable for the ODE $y' = f(t,y)$ as long as $\frac{\partial f}{\partial y}$ is in that range of $λ$ (and $f$ is smooth enough). We won’t prove this theorem here.

This statement has intuitive appeal since $\partial f/\partial y$ at a given point determines the behavior of the linearized equation, and one might imagine that applying stability analysis to the linearized equation at a point would give relevant information about the performance of the numerical method near that point, but I'm left wanting a more detailed discussion of this and proof.

Questions.

1. Where can I find a more detailed discussion of that second quoted statement?
2. In general, is linear stability analysis considered relevant to nonlinear systems because one can prove statements similar to that second quoted statement to the effect of "if the numerical method is stable for the linearized equation at every point, then it will be stable for the full, non-linear equation?"

References appreciated.