---
tag: DG7DZ37217
kind: definition
name: Oscillation theory
msc: [34C10]
defines: ZH2S50NSZ6
uses: [0QQRP17Z7S, 07VX34QXAV]
sources:
  - url: https://en.wikipedia.org/wiki/Oscillation_theory
---

## Definition

Let $t_0 \in \R$ and let $q \colon [t_0, \infty) \to \R$ be continuous.
Consider the [[linear differential equation|0QQRP17Z7S]]
$$y'' + q(t) y = 0 \quad \text{on } [t_0, \infty).$$
A real [[solution|07VX34QXAV]] $y$ that is not identically zero is oscillatory if it has arbitrarily large zeros.
That is, for each $T \ge t_0$ there is $t > T$ with $y(t) = 0$.
Otherwise $y$ is nonoscillatory.
The equation is oscillatory if it has an oscillatory solution, and nonoscillatory otherwise.
Oscillation theory is the study of the zeros of solutions of such equations and their generalisations.
It looks in particular for criteria for oscillation in terms of the coefficients.

## Remarks

By the Sturm separation theorem, if one nonzero solution is oscillatory then every nonzero solution is oscillatory.
So the equation is oscillatory exactly when all its nonzero solutions are.
Example: $y'' + y = 0$ is oscillatory, since $y = \sin t$ vanishes at $t = k\pi$ for every integer $k$.
Non-example: for $y'' = 0$ every nonzero solution $y = A t + B$ has at most one zero, so the equation is nonoscillatory.
The Euler equation $y'' + \frac{c}{t^2} y = 0$ on $[1, \infty)$ is oscillatory exactly when $c > \frac{1}{4}$.
A classical criterion says that the equation is oscillatory if $\int_{t_0}^\infty q(t) \, dt = +\infty$.

## Explanations

### L1 human

A solution oscillates if its graph keeps crossing the axis forever.
Oscillation theory looks for conditions on the equation that force, or forbid, this behaviour.