When it comes to differential equations, I notice that "Weakly Nonlinear" seems to be code for 'Relatively well behaved, easy to qualitatively predict, even if the solution requires an obscure special function or doesn't exist in closed form at all', whereas "Strongly Nonlinear" seems to mean 'Complicated, prone to chaos, don't even try finding a closed-form expression'.
Approaching this naively, if a differential equation has a nonlinear term in it, it is nonlinear. This is all I know from my limited maths education.
Do the terms 'Strong' and 'Weak' have a quantitative basis? Is it possible to inspect the differential equation and evaluate how weakly or strongly nonlinear the system is?
Or is 'Strong' and "Weak' nonlinearity just a qualitative descriptor of its behaviour and relative stability?
I have only a remark, but the standard comment above does not allow such a long text.
I have heard about strong or weak nonlinearity only in the qualitative (and often very subjective) way. However, what comes to my mind is for example the Kunkel path-following method (similar to the more popular arc-length method), which has adaptive step adjustment and for this purpose nonlinearity of the problem at the current step is measured. It is called $nonlinearity~index$ (Nichtlinearitätsindex) and it is defined in the step $t$ by \begin{align} ^tL=\frac{||^tu-^t\!\!u^{\text{1.it.}}||}{||^tu^{\text{1.it.}}-^{t-1}\!u||} \end{align} where $u^{t-1}$ is a value from the previous step, $u^t$ is the current iterate and $u^{t,\text{1.it.}}$ is the first iteration. Taken from the disseration of Eckart Resch, Zur numerischen Simulation von Holzverbindungen, 2008, pp.129.