Classification and integration of four-dimensional dynamical systems admitting non-linear superposition

Ibragimov, Nail H.; Gainetdinova, Aliya

doi:10.1016/j.ijnonlinmec.2017.01.008

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…Since its original formulation by Lie [23], nonautonomous first-order systems of ordinary differential equations admitting a nonlinear superposition rule, the so-called Lie systems, have been studied extensively (see [11,13,16,20,27,28,31,32] and references therein). The Lie theorem [14,23] states that every system of first-order differential equations is a Lie system if and only if it can be described as a curve in a finite-dimensional Lie algebra of vector fields, a referred to as Vessiot-Guldberg Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

Ballesteros

Campoamor-Stursberg

Fernández‐Saiz

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

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Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems [4], a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to sl(2) is proposed. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of sl(2) Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations. 1

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Section: Introductionmentioning

confidence: 99%

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

Ballesteros

Campoamor-Stursberg

Fernández‐Saiz

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…Our novel approach is presented in the next section, where the basics of LH systems and Poisson-Hopf algebras are recalled (for details on the general theory of Lie and LH systems, the reader is referred to [1,4,5,6,7,15,19,20,21,22,23,24,25,26,27,28]). To illustrate this construction, we consider in section 3 the Poisson-Hopf algebra analogue of the so-called non-standard quantum deformation of sl(2) [29,30,31,32] together with its deformed Casimir invariant.…”

Section: Introductionmentioning

confidence: 99%

Poisson–Hopf algebra deformations of Lie–Hamilton systems

Ballesteros¹,

Campoamor-Stursberg²,

Fernández‐Saiz³

et al. 2018

J. Phys. A: Math. Theor.

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Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new positiondependent mass oscillator system with a time-dependent frequency.

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Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

Campoamor-Stursberg

2018

Journal of Mathematical Physics

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A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot–Guldberg–Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot–Guldberg–Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

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Classification and integration of four-dimensional dynamical systems admitting non-linear superposition

Cited by 3 publications

References 8 publications

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

Poisson–Hopf algebra deformations of Lie–Hamilton systems

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

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