Recent Applications of the Theory of Lie Systems in Ermakov Systems

Abstract

Abstract. We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

“…The substitutionẋ = p further reduces Equation (11) to a first-order equation, recovering the well-known property that autonomous second-order ODEs admit a superposition principle. The same conclusion holds for autonomous SODE Lie systems in higher dimensions [5,13].…”

“…Among the low dimensional Lie algebras, the case r 2 sl (2, R) plays a special role, as it is related to various of the most relevant and best studied cases of SODE Lie systems (see, e.g., [2,4,5] and the references therein).…”

“…corresponding to the well-known generalizations of Ermakov systems (see, e.g., [5,17] and the references therein). We remark that a slight variation in the preceding argumentation also allows us to obtain SODE Lie systems having the Lie algebra r 5 as the Vessiot-Guldberg-Lie algebra.…”

“…For a long time considered as a particular technique for differential equations, the importance of Lie systems in physical applications, particularly in the context of integrable systems, as well as in control theory, has motivated extensive studies on the subject in the last few decades (see, e.g., [2,3]) that have led to natural generalizations of the notion of Lie systems [4][5][6]. These generalizations of the classical analytical formulation allow an elegant and effective description of Lie systems in terms of distributions in the sense of Fröbenius and the theory of fiber bundles, hence offering a much wider spectrum of applications, as well as their adaptation to quantum systems (the excellent treatise on the geometrical foundations of Lie systems [2] contains an extensive and updated list of references enumerating these applications).…”

Limit cycles for a class of second order differential equations

“…In the last years many people have retook their study, see for instance [2,4,5]. Since the classical Ermakov system has the form…”