I have been reading Strogatz's book on nonlinear differential equations, as well as watching Robert Ghrist's videos on nonlinear ODEs and discrete dynamical systems. I am just doing this as a self study.
I am understanding the math just fine, in the sense that I understand the qualitative methods, the analysis of fixed points and bifurcations, etc. But what I don't quite understand is some of the physical intuition behind the nonlinear terms in the models.
Let me give a couple of examples.
The normal form for a saddle node bifurcation is:
$$ \dot{x} = r + x^2 $$
and the normal form for a supercritical pitchfork bifurcation is:
$$ \dot{x} = rx - x^3 $$
I am just not clear how to interpret an $x^2$ term or $x^3$ term in some physical context. For the quadratic term, I can see a connection with the logistic(Verhulst) equation : $\dot{x} = rx(1 - x)$. So in that case, I get a stable fixed point at the value $1$ and an unstable fixed point at $0$. So the physical intution is that the rate of growth is proportional to that current state. And that as the population approaches the carrying capacity, then growth slows. Another way to think about this is as a mass action term, where instead of two different populations, the effect is relative to the population and itself.
But quadratic terms are used in other contexts too, and I am not clear on what is physically going on in those cases. For example, Strogatz cites the Spruce Budworm problem in 1-dimension. In that case, the predation term for the budworms by birds is defined as $\frac{Bx^2}{A^2 + x^2}$. Now I can plot this function on wolfram alpha and see what it looks like, but I still don't have a sense of why the authors of that paper use the quadratic terms versus a logistic function like $p(x) = \frac{1}{1 + e^{-kx}}$, which has a similar shape though slightly different slope. I can't think of a mass-action story to explain the form of this quadratic predation term.
For the cubic term $x^3$ I really have no idea what is happening there. It seems to deal with systems that have a sort of right and left symmetry, like a beam that can buckle in either direction. But outside of a beam buckling, I am not sure of what context a cubic term might be used?
Any suggestions or good examples of model intuition about these terms would really be helpful.