A new Lie systems approach to second-order Riccati equations

TL;DR

This paper introduces a Lie systems-based method for analyzing second-order Riccati equations, enabling their reduction to first-order Riccati equations via Legendre transforms, and providing new superposition rules.

Contribution

It presents a novel approach connecting second-order Riccati equations with Lie systems, facilitating their integration and analysis.

Findings

Mapped second-order Riccati equations to Lie systems

Developed new superposition rules for these equations

Reduced their integration to solving first-order Riccati equations

Abstract

This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre transforms. This result allows us to construct new superposition rules for studying second-order Riccati equations and to reduce their integration to solving (first-order) Riccati equations.