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Benito Hernández-Bermejo

Publications and source records attributed to Benito Hernández-Bermejo.

41 records · Page 3Linked to original sources
arXiv2019

Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

The determination of solutions of the Jacobi partial differential equations (PDEs) for finite-dimensional Poisson systems is considered. In particular, a novel procedure for the construction of solution families is developed. Such a procedure is based on the use of time reparametrizations preserving the existence of a Poisson structure. As a result, a method which is valid for arbitrary values of the dimension and the rank of the Poisson structure under consideration is obtained. In this article two main families of time reprametrizations of this kind are characterized. In addition, these results lead to a novel application which is also developed, namely the global and constructive determination of the Darboux canonical form for Poisson systems of arbitrary dimension and rank two, thus improving the local result provided by Darboux theorem for such a case.

math-ph↗
arXiv2019

New global solutions of the Jacobi partial differential equations

A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. This family is mathematically remarkable, as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves. This kind of Poisson structure (termed distinguished solutions or D-solutions) is defined for every nontrivial combination of values of the dimension and the rank, and is also determined in terms of functions of arbitrary nonlinearity, properties usually not present simultaneously in the already known solution families. In addition, D-solutions display several properties allowing the generation of an infinity of D-solutions from a given one, which is an uncommon feature in the framework of the Jacobi equations. Furthermore, a special family of D-solutions complying with the previous requirements is constructively characterized and analyzed. Examples are discussed focusing on physical implications and including an application for the global construction of the Darboux canonical form.

math-ph↗
arXiv2019

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method, both the Darboux coordinates and the Casimir invariants arise simultaneously as the outcome of the reduction algorithm. In fact, the congruence algorithm proceeds only in terms of matrix-algebraic transformations and direct quadratures, thus avoiding the need of previously integrating a system of PDEs and therefore improving previously known approaches. Physical examples illustrating different aspects of the theory are provided.

math-ph↗
arXiv2018

Equivalence between nonlinear dynamical systems and urn processes

An equivalence is shown between a large class of deterministic dynamical systems and a class of stochastic processes, the balanced urn processes. These dynamical systems are governed by quasi-polynomial differential systems that are widely used in mathematical modeling while urn processes are actively studied in combinatorics and probability theory. The presented equivalence extends a theorem by Flajolet et al. (Flajolet, Dumas and Puyhaubert Discr. Math. Theor. Comp. Sc. AG - 2006, DMTCS Proceedings) already establishing an isomorphism between urn processes and a particular class of differential systems with monomial vector fields. The present result is based on the fact that such monomial differential systems are canonical forms for more general dynamical systems.

nlin.CD↗