http://hdl.handle.net/2117/5344 2013-05-25T16:29:02Z http://hdl.handle.net/2117/16795 Title: Periodic orbits of integrable birational maps on the plane: blending dynamics and algebraic geometry, the Lyness' case Authors: Bastien, Guy; Mañosa Fernández, Víctor; Rogalski, Marc Abstract: Contingut del Pòster presentat al congrés New Trends in Dynamical Systems 2012-10-24T16:10:14Z http://hdl.handle.net/2117/15947 Title: Global periodicity conditions for maps and recurrences via Normal Forms Authors: Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor Abstract: We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences 2012-05-30T14:50:00Z http://hdl.handle.net/2117/14440 Title: On periodic solutions of 2-periodic Lyness difference equations Authors: Bastien, Guy; Mañosa Fernández, Víctor; Rogalski, Marc Abstract: We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears. Description: Preprint 2012-01-10T08:45:03Z http://hdl.handle.net/2117/14345 Title: Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version) Authors: Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor Abstract: This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features. Description: Preprint. Versió revisada i augmentada d'un anterior report homònim. 2011-12-29T09:24:44Z http://hdl.handle.net/2117/10770 Title: Integrability and non-integrability of periodic non-autonomous Lyness recurrences Authors: Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor Abstract: This paper studies non-autonomous Lyness type recurrences of the form x_{n+2}=(a_n+x_n)/x_{n+1}, where a_n is a k-periodic sequence of positive numbers with prime period k. We show that for the cases k in {1,2,3,6} the behavior of the sequence x_n is simple(integrable) while for the remaining cases satisfying k not a multiple of 5 this behavior can be much more complicated(chaotic). The cases k multiple of 5 are studied separately. Description: Preprint arXiv:1012.4925 2010-12-28T08:59:34Z http://hdl.handle.net/2117/7135 Title: Rational periodic sequences for the Lyness recurrence Authors: Gasull Embid, Armengol; Mañosa Fernández, Víctor; Xarles Ribas, Xavier Abstract: Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves. 2010-05-06T12:34:53Z http://hdl.handle.net/2117/6893 Title: On two and three periodic Lyness difference equations Authors: Cima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, Víctor Abstract: We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions x_1,x_2 are as well positive. We also show an interesting phenomenon of the discrete dynamical systems associated to some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behavior does not appear for the autonomous Lyness difference equations. 2010-04-09T11:12:00Z