# On purely nonlinear oscillators generalizing an isotonic potential

## TL;DR

This paper introduces a nonlinear generalization of the isotonic oscillator, constructing a symmetric potential with the same period function as the original asymmetric potential, expressed via hypergeometric functions.

## Contribution

It develops a symmetric potential generalizing the isotonic oscillator, with a period function dependent on amplitude and expressed through hypergeometric functions.

## Key findings

- Period function is amplitude-dependent.
- Reduces to 2π when α=1.
- Provides a hypergeometric function expression for the period.

## Abstract

In this paper we consider a nonlinear generalization of the isotonic oscillator in the same spirit as one considers the generalization of the harmonic oscillator with a truly nonlinear restoring force. The corresponding potential being asymmetric we invoke the symmetrization principle and construct a symmetric potential in which the period function has the same value as in the original asymmetric potential. The period function is amplitude dependent and expressible in terms of the hypergeometric function and reduces to $2\pi$ when $\alpha=1$, i.e., corresponding to the special case of an isotonic oscillator.