Periods of solutions of periodic differential equations

Smooth non-autonomous T-periodic differential equations x'(t)=f(t,x(t)) defined in \R\K^n, where \K is \R or \C and n 2 can have periodic solutions with any arbitrary period~S. We show that this is not the case when n=1. We prove that in the real C^1-setting the period of a non-constant periodic solution of the scalar differential equation is a divisor of the period of the equation, that is T/S\N. Moreover, we characterize the structure of the set of the periods of all the periodic solutions of a given equation. We also prove similar results in the one-dimensional holomorphic setting. In this situation the period of any non-constant periodic solution is commensurable with the period of the equation, that is T/S\Q

A polynomial class of Markus-Yamabe counterexamples

In the paper [CEGHM] a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension n ≥ 3 are given. In the present paper we explain a way for obtaining a family of polynomial counterexamples containing the above ones. Finally we study the global dynamics of the examples given in [CEGHM]

Dynamical classification of a family of birational maps of C2 via algebraic entropy

This work dynamically classifies a 9-parametric family of birational maps f: C→ C. From the sequence of the degrees d of the iterates of f, we find the dynamical degree δ(f) of f. We identify when d grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim (Mich Math J 54:647-670, 2006) as one of its subfamilies

Dynamics of the third order Lyness' difference equation

"Vegeu el resum a l'inici del document del fitxer adjunt"

Periodic orbits in complex Abel equation

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt = z 3 +B(t)z 2 +C(t)z, where B and C are smooth, 2π−periodic complex valuated functions, t ∈ R and z ∈ C. The first one is that there is no upper bound for its number of isolated 2π−periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π−periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt = A(t)z 3 + B(t)z 2 studied in the literature, where the center variety is located in a finite number of connected components

Invariant fibrations for some birational maps of ℂ2

In this article, we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them

Global eriodicity and complete integrability of discrete dynamical systems

Vegeu el resum a l'inici del document del fitxer adjun

A note on the Lyapunov and period constants

It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations

Zero entropy for some birational maps of C²

In this study, we consider a special case of the family of birational maps f:C² → C² , which were dynamically classified by [13]. We identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations. In particular, we highlight all of the integrable and periodic mappings

Pointwise periodic maps with quantized first integrals

We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically