相关论文: Mixed superposition rules and the Riccati hierarch…

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions…

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…

Mixed superposition rules are, in short, a method to describe the general solutions of a time-dependent system of first-order differential equations, a so-called Lie system, in terms of particular solutions of other ones. This article is…

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this…

The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…

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…

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use…

A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…

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…

This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution…

A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an…

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the…

We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a…

The theory of superposition rules for solutions of a Lie system of first-order differential equations is extended to deal with analogous systems of second-order and the theory is illustrated with the very rich example of Ermakov-like…

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei-Norman method is applied to obtain the associated…

It is proved that the members of the Riccati hierarchy, the so-called Riccati chain equations, can be considered as particular cases of projective Riccati equations, which greatly simplifies the study of the Riccati hierarchy. This also…

The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of…

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express…

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…

The $k$-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the $k$-symplectic structures to investigate a type of systems of…