# Hypergeometric First Integrals of the Duffing and van der Pol Oscillators

## TL;DR

This paper demonstrates the existence of hypergeometric first integrals for the Duffing and van der Pol oscillators, revealing new non-analytic integrals expressed via hypergeometric functions and providing a general criterion for such phenomena.

## Contribution

It introduces hypergeometric first integrals for these oscillators and formulates a criterion for their existence in general dynamical systems.

## Key findings

- Existence of Liouvillian first integrals expressed through hypergeometric functions.
- These integrals are neither analytic nor algebraic.
- A general criterion for hypergeometric integrals in dynamical systems.

## Abstract

The autonomous Duffing oscillator, and its van der Pol modification, are known to admit time-dependent first integrals for specific values of parameters. This corresponds to the existence of Darboux polynomials, and in fact more can be shown: that there exist Liouvillian first integrals which do not depend on time. They can be expressed in terms of the Gauss and Kummer hypergeometric functions, and are neither analytic, algebraic nor meromorphic. A criterion for this to happen in a general dynamical system is formulated as well.