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http://hdl.handle.net/2117/5344
Wed, 30 Jul 2014 17:29:17 GMT2014-07-30T17:29:17Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoPeriodic orbits of integrable birational maps on the plane: blending dynamics and algebraic geometry, the Lyness' case
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 SystemsWed, 24 Oct 2012 16:10:14 GMThttp://hdl.handle.net/2117/167952012-10-24T16:10:14ZBastien, Guy; Mañosa Fernández, Víctor; Rogalski, MarcnoPeriodic orbits; Birational maps; Integrability; Nonlinear dynamics; Elliptic curves.Contingut del Pòster presentat al congrés New Trends in Dynamical SystemsGlobal periodicity conditions for maps and recurrences via Normal Forms
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 recurrencesWed, 30 May 2012 14:50:00 GMThttp://hdl.handle.net/2117/159472012-05-30T14:50:00ZCima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, VíctornoPeriodic maps, Linearization, Normal Forms, Rational parametrizations, Globally periodic recurrences, Lyness recurrencesWe 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 recurrencesOn periodic solutions of 2-periodic Lyness difference equations
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: PreprintTue, 10 Jan 2012 08:45:03 GMThttp://hdl.handle.net/2117/144402012-01-10T08:45:03ZBastien, Guy; Mañosa Fernández, Víctor; Rogalski, MarcnoDifference equations with periodic coefficients, Elliptic curves, Lyness' type equations, QRT maps, Rotation number, Periodic orbitsWe 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.Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)
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.Thu, 29 Dec 2011 09:24:44 GMThttp://hdl.handle.net/2117/143452011-12-29T09:24:44ZCima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, VíctornoIntegrability and non-integrability of discrete systems, Numerical chaos, Periodic difference equations, QRT maps, Rational and meromorphic first integralsThis 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.Integrability and non-integrability of periodic non-autonomous Lyness recurrences
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:1012.4925Tue, 28 Dec 2010 08:59:34 GMThttp://hdl.handle.net/2117/107702010-12-28T08:59:34ZCima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, VíctornoPeriodic difference equations, Integrability, Non-integrability, Meromorphic first integrals, ChaosThis 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.Rational periodic sequences for the Lyness recurrence
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.Thu, 06 May 2010 12:34:53 GMThttp://hdl.handle.net/2117/71352010-05-06T12:34:53ZGasull Embid, Armengol; Mañosa Fernández, Víctor; Xarles Ribas, XaviernoLyness difference equations, Rational points over elliptic curves, Periodic points, Universal family of elliptic curvesConsider 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.On two and three periodic Lyness difference equations
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.Fri, 09 Apr 2010 11:12:00 GMThttp://hdl.handle.net/2117/68932010-04-09T11:12:00ZCima Mollet, Anna; Gasull Embid, Armengol; Mañosa Fernández, VíctornoDifference equations with periodic coefficients, Circle maps, Rotation numberWe 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.