Lie systems: Theory, generalisations, and applications

Abstract

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of mathematics and physics. These facts, together with the authors' recent findings in the theory of Lie systems, led them to write this essay, which aims to describe the new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications. © 2007-2012 by IMPAN. All rights reserved.