Mixed superposition rules and the Riccati hierarchy

TL;DR

This paper generalizes the Lie-Scheffers Theorem to characterize systems with mixed superposition rules, showing they are exactly Lie systems, and applies this to analyze the Riccati hierarchy more efficiently.

Contribution

It extends the Lie-Scheffers Theorem to mixed superposition rules, providing a new method to identify Lie systems and simplifying analysis of the Riccati hierarchy.

Findings

Systems with mixed superposition rules are exactly Lie systems.

The generalized theorem simplifies the study of the Riccati hierarchy.

New tools for identifying Lie systems are introduced.

Abstract

Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.