The invisible heartbeat. The beauty and soul of mathematics

These notes contain a summarised version of the contents of the lectures “The invisible heartbeat. The beauty and soul of mathematics” and “The borders of our prejudices”, given at the Casa de Cultura in Girona in October 2015 and September 2018 respectively, within the “Contemporalia” cycle organized by the Lluís A. Santaló Chair of Mathematical Applications of the University of GironaWe will look at mathematical aesthetic experience from a point of view close to the one of analytical psychology. We will analyse the myth that the universe is a harmonic entity that can be described through mathematics, giving it its beauty, and study the unconscious remnant of this archetypal idea in contemporary science. Mathematical beauty appears, finally, as a link between the Archetype of the Cosmos and the wholeness of the psyche. These notes contain a summarised version of the contents of the lectures “The invisible heartbeat. The beauty and soul of mathematics” and “The borders of our prejudices”, given at the Casa de Cultura in Girona in October 2015 and September 2018 respectively, within the “Contemporalia” cycle organized by the Lluís A. Santaló Chair of Mathematical Applications of the University of Girona.Preprin

Singular solutions for a class of traveling wave equations arising in hydrodynamics

We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u + \frac{1}{2}\dot{u}^2 + F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.Comment: 24 pages, 5 figure

Lie symmetries of birational maps preserving genus 0 fibrations

Preprint.We prove that any planar birational integrable map, which preserves a fibration given by genus $0$ curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods of the considered maps.Preprin

Singular solutions for a class of traveling wave equations arising in hydrodynamics

PreprintWe give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u + \frac{1}{2}\dot{u}^2 + F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.Preprin

Basin of attraction of triangular maps with applications

We consider some planar triangular maps. These maps preserve certain fibration of the plane. We assume that there exists an invariant attracting fiber and we study the limit dynamics of those points in the basin of attraction of this invariant fiber, assuming that either it contains a global attractor, or it is filled by fixed or 2-periodic points. Finally, we apply our results to a variety of examples, from particular cases of triangular systems to some planar quasi-homogeneous maps, and some multiplicative and additive difference equations, as well.Comment: 1 figur

Non-integrability of measure preserving maps via Lie symmetries

We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.Comment: 25 page

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.Comment: 21 page

El batec invisible. La bellesa i l’ànima de les matemàtiques

Aquestes notes contenen una versió resumida dels continguts de les conferencies “El batec invisible. La bellesa i l’ànima de les matemàtiques” i “Les fronteres dels nostres prejudicis”, pronunciades a la Casa de Cultura de Girona l’octubre de 2015 i el setembre de 2018 respectivament, dins del cicle “Contemporàlia” organitzat per la Càtedra Lluís Santaló d’Aplicacions de la Matemàtica.Ens aproximem a l’experiència estètica matemàtica des d’un punt de vista proper al de la psicologia analítica. Analitzem el mite segons el qual l’univers és un ens harmònic que pot ser descrit mitjançant les matemàtiques, les quals li atorguen la seva bellesa, i estudiem el romanent inconscient d’aquesta idea arquetípica en la ciència contemporània. La bellesa matemàtica es presenta, finalment, com a vincle entre l’Arquetip del Cosmos i el de la totalitat de la psique. Aquestes notes contenen una versió resumida dels continguts de les conferencies “El batec invisible. La bellesa i l’ànima de les matemàtiques” i “Les fronteres dels nostres prejudicis”, pronunciades a la Casa de Cultura de Girona l’octubre de 2015 i el setembre de 2018 respectivament, dins del cicle “Contemporàlia” organitzat per la Càtedra Lluís Santaló d’Aplicacions de la Matemàtica.Preprin

Periodic orbits of planar integrable birational maps

Conferència convidada al congrés NOMA'13.Preprin

Minor loops of the Dahl and LuGre models

PreprintHysteresis is a special type of behavior encountered in physical systems: in a hysteretic system, when the input is periodic and varies slowly, the steady-state part of the output-versus-input graph becomes a loop called hysteresis loop. In the presence of perturbed inputs or noise, this hysteresis loop presents small lobes called minor loops that are located inside a larger curve called major loop. The study of minor loops is being increasingly popular since it leads to a quantification of the loss of energy due to the noise. The aim of the present paper is to give an explicit analytic expression of the minor loops of the LuGre and the Dahl models of dynamic dry friction.Preprin