Non-autonomous hamiltonian systems and Morales-Ramis theory

In this paper we present an approach towards the comprehensive analysis of the non-integrability of differential equations in the form $\ddot x=f(x,t)$ which is analogous to Hamiltonian systems with $1+1/2$ degree of freedom. In particular, we analyze the non-integrability of some important families of differential equations such as Painlevé II, Sitnikov and Hill-Schrödinger equation. We emphasize in Painlevé II, showing its non-integrability through three different Hamiltonian systems, and also in Sitnikov in which two different version including numerical results are shown. The main tool to study the non-integrability of these kind of Hamiltonian systems is Morales-Ramis theory. This paper is a very slight improvement of the talk with the same title delivered by the author in SIAM Conference on Applications of Dynamical Systems 2007

An application of functional analysis in a predator - prey system

In this paper I want to show some advantages of the Geometric Iteration to a possible study of a predator-prey system. The Geometric Iteration presents characteristics such as continuity, derivability and integrability which could allow to analyze when a predator can still reach it is prey even when the prey has an initial advantage.Peer Reviewe

La teoría de Morales-Ramis y el algoritmo de Kovacic

La teor\'\ı a de Morales–Ramis es la teor\'\ı a de Galois en el contexto de los sistemas din\'amicos y relaciona dos tipos diferentes de integrabilidad: integrabilidad en el sentido de Liouville de un sistema hamiltoniano e integrabilidad en el sentido de la teor\'\ı a de Galois diferencial de una ecuaci\'on diferencial. En este art\'\i culo se presentan algunas aplicaciones de la teor\'\i a de Morales–Ramis en problemas de no integrabilidad de sistemas hamiltonianos cuya ecuaci\'on variacional normal a lo largo de una curva integral particular es una ecuaci\'on diferencial lineal de segundo orden con coeficientes funciones racionales. La integrabilidad de la ecuaci\'on variacional normal es analizada mediante el algoritmo de Kovacic.Peer Reviewe

Un modelo de captura con las torres y la exponencial iterada

En este artículo se trata un caso especial de un sistema depredador-presa, donde el tamaño de los pasos (o saltos) de ambas especies crece como exponenciales iteradas. Se supone que el depredador da a su presa una ventaja inicial antes de empezar la cacería: ¿es posible que la presa escape del depredador? Se darán condiciones sobre el tamaño de los pasos (o saltos) de ambas especies para determinar si la presa escapa a su depredador.  In this paper it is assumed an special case of a predator-prey system, in which the length of step of both species grow as iterated exponentials. It is supposed that the predator give to its prey an initial advantage before of to start the chase: could the prey scape to its predator? Conditions about the size of jumps or strides of both species to determine if the prey can scape to its predator will be given

Genealogía de permutaciones simples de orden una potencia de dos

The aim of this paper is to show some properties of simple permutations with order a power of two and to give a combinatorial formula to determine its genealogy involving two new operations: the \textit{pasting} operation and \textit{reversing}. Simple permutations are very important because corresponds to primary orbits or minimal orbits and in particular, simple permutations with order a power of two are related with the right side in the Sharkovskii's order

Un modelo de captura con las torres y la exponencial iterada

En este artículo se trata un caso especial de un sistema depredador-presa, donde el tamaño de los pasos (o saltos) de ambas especies crece como exponenciales iteradas. Se supone que el depredador da a su presa una ventaja inicial antes de empezar la cacería: ¿es posible que la presa escape del depredador? Se darán condiciones sobre el tamaño de los pasos (o saltos) de ambas especies para determinar si la presa escapa a su depredador.

Un modelo de captura con las torres y la exponencial iterada

En este artículo se trata un caso especial de un sistema depredador-presa, donde el tamaño de los pasos (o saltos) de ambas especies crece como exponenciales iteradas. Se supone que el depredador da a su presa una ventaja inicial antes de empezar la cacería: ¿es posible que la presa escape del depredador? Se darán condiciones sobre el tamaño de los pasos (o saltos) de ambas especies para determinar si la presa escapa a su depredador.  In this paper it is assumed an special case of a predator-prey system, in which the length of step of both species grow as iterated exponentials. It is supposed that the predator give to its prey an initial advantage before of to start the chase: could the prey scape to its predator? Conditions about the size of jumps or strides of both species to determine if the prey can scape to its predator will be given

Darboux integrals for Schrödinger planar vector fields via Darboux transformations

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the “invariance” of the elements of the “Darboux Theory of Integrability”. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Free particle, square well, harmonic oscillator, three dimensional harmonic oscillator and Coulomb potential, are presented as natural examples coming from supersymmetric quantum mechanics.Preprin