189 research outputs found
On the set of periods of the 2-periodic Lyness' equation
Publicació amb motiu de la International Conference on Difference Equations and Applications (July 22-27, 2012, Barcelona, Spain) amb el títol Difference Equations, Discrete Dynamical Systems and ApplicationsWe study the periodic solutions of the non-autonomous periodic Lyness' recurrence u = (a + u )/u, where {a} 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) ≠ (1, 1), then there exists a value p(a, b) such that for any p > p(a, b) there exist continua of initial conditions giving rise to 2p-periodic sequences. (2) The set of minimal periods arising when (a, b) ∈ (0,∞) and positive initial conditions are considered, contains all the even numbers except 4, 6, 8, 12 and 20. If a ≠ b, then it does not appear any odd period, except 1
Non-integrability of measure preserving maps via Lie symmetries
We consider the problem of characterizing, for certain natural number ,
the local -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
Studying discrete dynamical systems trough differential equations
In this paper we consider dynamical systems generated by a diffeomorphism F
defined on U an open subset of R^n, and give conditions over F which imply that
their dynamics can be understood by studying the flow of an associated
differential equation, also defined on U. In particular the case
where F has n-1 functionally independent first integrals is considered. In this
case X is constructed by imposing that it shares with the same set of first
integrals and that the functional equation for
x in U has some non-zero solution. Several examples for n=2,3 are presented,
most of them coming from several well-known difference equations.Comment: 22 pages; 3 Figure
Some properties of the k-dimensional Lyness' map
This paper is devoted to study some properties of the k-dimensional Lyness'
map. Our main result presentes a rational vector field that gives a Lie
symmetry for F. This vector field is used, for k less or equal to 5 to give
information about the nature of the invariant sets under F. When k is odd, we
also present a new (as far as we know) first integral for F^2 which allows to
deduce in a very simple way several properties of the dynamical system
generated by F. In particular for this case we prove that, except on a given
codimension one algebraic set, none of the positive initial conditions can be a
periodic point of odd period.Comment: 22 pages; 3 figure
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
On the set of periods of the 2-periodic Lyness’ Equation
PreprintWe 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.Preprin
On two and three periodic Lyness difference equations
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.Preprin
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