Francesc Mañosas

A Chebyshev criterion for Abelian integrals

We present a criterion that provides an easy sufficient condition in order for a collection of Abelian integrals to have the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.

The focus-centre problem for a type of degenerate system

We consider differential systems in the plane defined by the sum of two homogeneous vector fields. We assume that the origin is a degenerate singular point for these differential systems. We characterize when the singular point is of focus-centre type in a generic case. The problem of its local stability is also considered. We compute the first generalized Lyapunov constant when some non-degeneracy conditions are assumed.

On Periodic Rational Difference Equations of Order k

This paper is devoted to the study of which rational difference equations of order k, with non negative coefficients, are periodic. Our main result is that for and for only the well known periodic difference equations and their natural extensions appear.

Canonical representatives for patterns of tree maps

Abstract We define a notion of pattern for finite invariant sets of continuous maps of finite trees. A pattern is essentially a homotopy class relative to the finite invariant set. Given such a pattern, we prove that the class of tree maps which exhibit this pattern admits a canonical representative, that is a tree and a continuous map on this tree, which satisfies several minimality properties. For instance, it minimizes topological entropy in its class and its dynamics are minimal in a sense to be defined. We also give a formula to compute the minimal topological entropy directly from the combinatorial data of the pattern. Finally we prove a characterization theorem for zero entropy patterns.

Lower bounds of the topological entropy for continuous maps of the circle of degree one

The authors give the best lower bound of the topological entropy of a continuous map of the circle of degree one, as a function of the rotation interval. Also, they obtain as a corollary the theorem of Ito (1982) which gives the best lower bound of the topological entropy depending on the set of periods.

A NOTE ON THE CRITICAL PERIODS OF POTENTIAL SYSTEMS

In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x,y) = (1/2)y2+V(x) where V(x) = (1/2)x2+(a/4)x4+(b/6)x6 with b ≠ 0. This family of differentia...

Minimizing topological entropy for continuous maps on graphs

We study the existence of models with minimal topological entropy among the class of all continuous maps from a given graph to itself with a fixed behavior on a given finite invariant set.

A polynomial class of Markus-Yamabe counterexamples

In the paper \cite{CEGHM} a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension

Isochronicity of a class of piecewise continuous oscillators

n\ge 3

A Poincaré-Hopf theorem for noncompact manifolds☆

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 \cite{CEGHM}.