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Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the lower order terms which really govern behaviour around the point of interest.

However, there are cases where we cannot throw out the higher order terms, because all the lower order terms are zero, and it is the higher order terms that are non-zero, thus governing behaviour.

I think one example of such a phenomenon is the 2D "elastic buckling beam" -- the beam (under the action of a force) can buckle either left or right, or simply be crushed straight down, and thus the "bifurcation diagram" for its behaviour has a so-called three-pronged "pitchfork bifurcation", corresponding to: buckling to the left, buckling to the right, and crushing. In the case of the elastic buckling beam, the ODE governing its behaviour has a Taylor series approximation where the order 0, 1 and 2 terms are zero, but the order 3 terms and above are non-zero. Thus, the ODE can roughly be written as a cubic curve, with at least 3 "solutions" corresponding to buckling to the left, right, and crushing.

What are some other physical examples corresponding to situations where only the higher order Taylor series terms survive?

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A function is also $\,f(z)=a\,z^{14}+b\,z^{15}+c\,z^{16}$ is just $z^{14}(a+b\,z+c\,z^2)$. And if you do a taylor expansion of $z^19+3z^6$ around $z_0=5$, you will find a linear term as well.

Regarding the physics aspect with quantities where the minimum matters: Take any partition function of the form $z$ times something, average a couple of quadratic quantities with it, and you're there.

Consider Planck's law... Planck showed that the spectral radiance of a body at absolute temperature is given by $$B(\nu) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\mathrm{B}T} - 1}\propto \nu^2+\dots$$

Or from it... the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time $j$ is directly proportional to the fourth power of the black body's thermodynamic temperature $$j \propto T^{4}.$$

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