A Guide to Lie Systems with Compatible Geometric Structures

Muñoz, Cristina Sardón; Lucas, Javier de; Araujo, Javier De Lucas

doi:10.1142/q0208

Cited by 25 publications

(102 citation statements)

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“…Let us provide a brief account of the theory of Lie-Hamilton systems needed to understand our work and to make our exposition almost self-contained. To highlight our main ideas and to avoid minor technical details, we assume, if not otherwise stated, that mathematical objects are smooth and well defined globally (see [1,7,9,17,25] for further details). Hereafter, N stands for a connected ndimensional manifold, and g denotes an abstract finite-dimensional Lie algebra.…”

Section: Fundamentalsmentioning

confidence: 99%

“…Each t-dependent vector field X on N amounts to a t-parametric family of standard vector fields on N of the form {X t : x ∈ N → X(t, x) ∈ T N } t∈R . We call smallest Lie algebra of X (also called minimal Lie algebra or irreducible Lie algebra in the literature [17]) the smallest Lie algebra (in the sense of inclusion) of vector fields, V X , containing {X t } t∈R . Let us denote by D V the generalised distribution on N spanned by the elements of a Lie algebra of vector fields V .…”

Section: Fundamentalsmentioning

confidence: 99%

“…A superposition rule [8,9,17,29] for a system X on N is a map Ψ : N m × N → N satisfying that the general solution, x(t), to X can be expressed as…”

Section: Fundamentalsmentioning

confidence: 99%

“…Although the term superposition rule has been used in the literature with different meanings and there exist different approaches to each notion, our definition in this paper is the predominant in the literature on nonlinear differential equations (cf. [7,9,17,18,24] and references therein).…”

Section: Fundamentalsmentioning

confidence: 99%

“…Every Lie system on R 2 can be endowed with a VG Lie algebra that is, around a generic point, locally diffeomorphic to one of the twenty-eight possible classes of VG Lie algebras on R 2 described in [12,15,17]. In particular, only twelve classes can be considered, again locally around a generic point, as VG Lie algebras of Hamiltonian vector fields relative to a symplectic form (see [6,17] and Table 1). Lie systems admitting a VG Lie algebra of Hamiltonian vector fields relative to a Poisson bivector are called Lie-Hamilton systems [6,10,17].…”

Section: Introductionmentioning

confidence: 99%

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Geometric Models for Lie–Hamilton Systems on ℝ2

Lange

Lucas

2019

Mathematics

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This paper provides a geometric description for Lie-Hamilton systems on R 2 with locally transitive Vessiot-Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie-Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie-Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie-Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie-Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

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Section: Fundamentalsmentioning

confidence: 99%

Section: Fundamentalsmentioning

confidence: 99%

“…A superposition rule [8,9,17,29] for a system X on N is a map Ψ : N m × N → N satisfying that the general solution, x(t), to X can be expressed as…”

Section: Fundamentalsmentioning

confidence: 99%

Section: Fundamentalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geometric Models for Lie–Hamilton Systems on ℝ2

Lange

Lucas

2019

Mathematics

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A generalization of a SIS epidemic model with fluctuations

Esen

Fernández‐Saiz

Sardón

et al. 2021

Math Methods in App Sciences

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In a recent paper (Nakamura and Martinez, 2019), the classical epidemic compartmental susceptible-infectious-susceptible (SIS) model has been upgraded to a form which permits fluctuations in terms of the mean and the variance of infected individuals. This novel model happens to admit a Hamiltonian realization involving a constant 𝜌 0 that is related to the reproduction number R 0 . In this work, we generalize 𝜌 0 as a t-dependent function to arrive at a non-autonomous system of differential equations that we call SISt model. We make use of the theory of Lie systems to provide a (nonlinear) superposition rule for the general solution of SISt model. To conclude, we present a quantum version of the same model, giving rise to a parametric family of SISt systems. We provide a superposition rule for the general solution of the quantum extension as well.

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