Are there periodic functions satisfying a quadratic differential equation?

Question

Question: Are there periodic functions satisfying a quadratic differential equation, as opposed to just linear or cubic?

Bonus question: Are there periodic functions satisfying differential equations which are polynomials of any degree $n$?

Background: I know that on the real line, any periodic function can be decomposed into a (possibly infinite) sum of sines and cosines via Fourier series, and that this technique is somewhat extensible to the complex plane (I think).

Likewise, cosine and sine can be defined as the solutions to a linear systems of ordinary differential equation (at least on the real line, I am not sure about the complex plane), see e.g here.

A natural generalization/extension of the cosine and sine would be to either consider (1) functions defining the trigonometry of more general conic sections than the circle or (2) functions which have more "advanced" periodicity properties, like double periodicity.

Functions which seem to satisfy both of these criteria would be the elliptic functions, because they are (1) the inverses of elliptic integrals, and (2) doubly periodic.

However, what is surprising to me is that the (complex) differential equation which they satisfy is cubic in the function and its first derivative, rather than quadratic. This seems to suggest a substantial jump in complexity from the trigonometric functions.

This question is motivated by Algebraic Geometry: A Problem Solving Approach, which mentions the elliptic functions in the context of cubic curves, but does not mention any special functions related to conic sections -- is this because there aren't any, or are they just somehow less important?

Attempt: Maybe the hyperbolic tangent function? I don't think that it's periodic, but it is related to the exponential, which is periodic. Also it seems to satisfy a quadratic differential equation.

Note: I tagged this (complex-analysis) because it seems to be about complex differential equations.

Answer 1

(I assume you are talking about the second-order ODEs for trig functions or (Jacobi) elliptic functions?)

The equation $d^2 x/dt^2 = x^2-x$, which is quadratic in $x(t)$, has (some) periodic solutions, as can be seen by writing it as $$dx/dt=y,\qquad dy/dt=x(x-1)$$ and sketching the phase portrait in the $xy$ plane. But if you try to integrate it explicitly, you get $(dx/dt)^2/2 = x^3/3-x^2/2+C$, which leads to an integral with a square root of a cubic polynomial, and thus to elliptic functions again.

Answer 2

Update: I just realized that this is probably a really dumb question, specifically, consider that $\frac{d}{dx}\sin x = \cos x$. Set $y = \sin x$. Then since $\sin ^2 x + \cos ^2 x =1$, we have that $$(y')^2 + y^2 =1 $$ which is exactly the sort of thing I was thinking about. I.e. the sinusoidal functions correspond to the unit circle exactly in the way that elliptic curves correspond to (Weierstrass) elliptic functions.

Likewise, if one sets $y = \sinh x$, then one has that $$ (y')^2 - y^2 =1$$ which means that the hyperbolic trigonometric functions (unsurprisingly) correspond to the hyperbola conic sections. Thus the "quadratic differential equation periodic functions" I was talking about were just the trigonometric functions after all.

If anyone can think of a "parabola" counterpart to the unit ellipse and unit hyperbola examples I have above I would greatly appreciate it.

EDIT: a similar question has been asked before on Math.SE, see for example: Do "Parabolic Trigonometric Functions" exist?, also Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

Also if anyone knows anything about whether or not there are corresponding special functions for all (smooth) algebraic curves, or if it is just the case for conic sections and elliptic curves, I would be both very interested and very grateful to know.

This might not be possible since if smooth quadratic curves correspond to singly periodic real-valued functions, and smooth cubic curves correspond to doubly periodic complex-valued functions, then smooth higher-degree curves might correspond to "n-th ly periodic" functions of even higher order division algebras -- however none exist that are fields. Also Jacobi proved that triply periodic functions and higher cannot exist, so my naive guess is that the answer is no.