# Non-integrability of the Huang--Li nonlinear financial model

## TL;DR

This paper proves that the Huang--Li nonlinear financial system, modeled by three differential equations, is not integrable in meromorphic functions using differential Galois theory.

## Contribution

It provides the first analytic proof of the non-integrability of the Huang--Li financial model through differential Galois group analysis.

## Key findings

- The system is non-integrable in meromorphic functions.
- Differential Galois theory confirms non-integrability.
- Analytic proof based on variational equations.

## Abstract

In this paper we consider Huang--Li nonlinear financial system recently studied in the literature. It has the form of three first order differential equations \[ \dot x=z+(y-a)x,\quad \dot y=1-b y-x^2,\quad \dot z=-x-c z, \] where $(a,b,c)$ are real positive parameters. We show that this system is not integrable in the class of functions meromorphic in variables $(x,y,z)$. We give an analytic proof of this fact analysing properties the of differential Galois group of variational equations along certain particular solutions of the system.