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On two and three periodic Lyness difference equations
(2009-12-26)
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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 ...
On the set of periods of the 2-periodic Lyness’ Equation
(2013-07-25)
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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 ...
Continua of periodic points for planar integrable rational maps.
(2015-11-30)
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We present three alternative methodologies to find continua of periodic points with a prescribed period for rational maps having rational first integrals. The first two have been already used for other authors and apply ...
Persistence of periodic traveling waves and Abelian integrals
(2021-03-10)
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It is well known that the existence of traveling wave solutions (TWS) for many partial differential equations (PDE) is a consequence of the fact that an associated planar ordinary differential equation (ODE) has certain ...
Bifurcation of 2-periodic orbits from non-hyperbolic fixed points
(2017-07-21)
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We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful ...
Non autonomous 2-periodic Gumovski-Mira difference equations
(2011-06-01)
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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 ...
Phase portraits of random planar homogeneous vector fields
(2019-11-26)
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We study the phase portraits with positive probability of random planar homogeneous vector fields of degree n. In particular, for n=1,2,3, we give a complete solution of the problem and, moreover, either we give the exact ...
Non-integrability of measure preserving maps via Lie symmetries
(2015-03-18)
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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 ...
The invisible heartbeat. The beauty and soul of mathematics
(2020-01-15)
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We 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, ...
Singular solutions for a class of traveling wave equations arising in hydrodynamics
(2015-02-19)
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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 ...