Reports de recerca
http://hdl.handle.net/2117/5340
Wed, 29 Jul 2015 03:15:39 GMT2015-07-29T03:15:39ZNon-integrability of measure preserving maps via Lie symmetries
http://hdl.handle.net/2117/26843
Non-integrability of measure preserving maps via Lie symmetries
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor
We consider the problem of characterizing, for certain natural
number m, the local 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.
Preprint.
Wed, 18 Mar 2015 00:00:00 GMThttp://hdl.handle.net/2117/268432015-03-18T00:00:00ZSingular solutions for a class of traveling wave equations arising in hydrodynamics
http://hdl.handle.net/2117/26450
Singular solutions for a class of traveling wave equations arising in hydrodynamics
Geyer, Anna; Mañosa Fernández, Víctor
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.
Preprint
Thu, 19 Feb 2015 00:00:00 GMThttp://hdl.handle.net/2117/264502015-02-19T00:00:00ZLie symmetries of birational maps preserving genus 0 fibrations
http://hdl.handle.net/2117/26449
Lie symmetries of birational maps preserving genus 0 fibrations
Llorens, Mireia; Mañosa Fernández, Víctor
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.
Preprint.
Fri, 13 Feb 2015 00:00:00 GMThttp://hdl.handle.net/2117/264492015-02-13T00:00:00ZPeriodic orbits of planar integrable birational maps
http://hdl.handle.net/2117/21748
Periodic orbits of planar integrable birational maps
Gálvez Carrillo, Maria Immaculada; Mañosa Fernández, Víctor
A birational planar map F possessing a rational ﬁrst integral preserves a
foliation of the plane given by algebraic curves which, if F is not globally periodic,
is given by a foliation of curves that have generically genus 0 or 1. In the genus 1
case, the group structure of the foliation characterizes the dynamics of any birational
map preserving it. We will see how to take advantage of this structure to ﬁnd periodic
orbits of such maps.
Fri, 14 Feb 2014 00:00:00 GMThttp://hdl.handle.net/2117/217482014-02-14T00:00:00ZBasin of attraction of triangular maps with applications
http://hdl.handle.net/2117/20131
Basin of attraction of triangular maps with applications
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor
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.
Preprint
Thu, 25 Jul 2013 00:00:00 GMThttp://hdl.handle.net/2117/201312013-07-25T00:00:00ZDifferent approaches to the global periodicity problem
http://hdl.handle.net/2117/20123
Different approaches to the global periodicity problem
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor; Mañosas, Francesc
t Let F be a real or complex n-dimensional map. It is said that F is globally
periodic if there exists some p ∈ N
+ such that Fp(x) = x for all x, where F
k = F ◦ F k−1, k ≥ 2. The minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say Fλ, a problem of current interest is to determine all the values of λ such that Fλ is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we
use to face this question, as well as some recent results that we have obtained. We
will focus on proving the equivalence of the problem with the complete integrability
of the dynamical system induced by the map F, and related issues; on the use of the
local linearization given by the Bochner Theorem; and on the use the Normal Form
theory. We also present some open questions in this setting.
Preprint
Thu, 25 Jul 2013 00:00:00 GMThttp://hdl.handle.net/2117/201232013-07-25T00:00:00ZOn the set of periods of the 2-periodic Lyness’ Equation
http://hdl.handle.net/2117/20122
On the set of periods of the 2-periodic Lyness’ Equation
Bastien, Guy; Mañosa Fernández, Víctor; Rogalski, Marc
We study the periodic solutions of the non–autonomous periodic Lyness’ recurrence un+2 = (an +un+1)=un, where fangn is a cycle with positive values a,b and with positive initial conditions. Among other methodological issues we give an outline of the proof of the following results: (1) If (a;b) 6= (1;1), then there exists a value p0(a;b) such that for any p > p0(a;b) there exist continua of initial conditions giving rise to 2p–periodic sequences. (2) The set of minimal periods arising when (a;b) 2 (0;¥) 2 and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a 6= b, then it does not appear any odd period, except 1.
Preprint
Thu, 25 Jul 2013 00:00:00 GMThttp://hdl.handle.net/2117/201222013-07-25T00:00:00ZPeriodic orbits of planar integrable birational maps
http://hdl.handle.net/2117/20107
Periodic orbits of planar integrable birational maps
Mañosa Fernández, Víctor
Conferència convidada al congrés NOMA'13.
Tue, 03 Sep 2013 00:00:00 GMThttp://hdl.handle.net/2117/201072013-09-03T00:00:00ZSpectral properties of the connectivity matrix and the SIS-epidemic threshold for mid-size metapopulations
http://hdl.handle.net/2117/19168
Spectral properties of the connectivity matrix and the SIS-epidemic threshold for mid-size metapopulations
Juher Barrot, David; Mañosa Fernández, Víctor
We consider the spread of an infectious disease on a heterogeneous metapopulation
deﬁned by any (correlated or uncorrelated) network. The infection evolves under transmission, recovering and migration mechanisms. We study some spectral properties of a connectivity matrix arising from the continuous-time equations of the model. In particular we show that the classical sufﬁcient condition of instability for the disease-free equilibrium, well known for the particular case of uncorrelated networks, works also for the general case. We give also an alternative condition that yields a more accurate estimation of the epidemic threshold for correlated (either assortative or dissortative) networks
Preprint version of the paper
Fri, 10 May 2013 00:00:00 GMThttp://hdl.handle.net/2117/191682013-05-10T00:00:00ZNon autonomous 2-periodic Gumovski-Mira difference equations
http://hdl.handle.net/2117/12720
Non autonomous 2-periodic Gumovski-Mira difference equations
Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor
We consider two types of non-autonomous 2-periodic Gumovski-Mira difference equations. We show that while the corresponding autonomous recurrences are conjugated, the behavior of the sequences generated by the 2-periodic ones di er dramatically: in one case the behavior of the sequences is simple (integrable) and in the other case it is much more complicated (chaotic). We also present a global study of the integrable case that includes which periods appear for the recurrence.
Wed, 01 Jun 2011 00:00:00 GMThttp://hdl.handle.net/2117/127202011-06-01T00:00:00Z