Types d'orbites et dynamique minimale pour les applications continues de graphesOrbit types and minimal dynamics for graph maps

We define the type of a periodic orbit of a graph map. We consider the class of ‘train-track’ representatives, that is, those graph maps which minimise the topological entropy of the topological representatives of a given free group endomorphism. We prove that each type of periodic orbit realised by an efficient representative is also realised by any representative of the same free group endomorphism. Moreover, the number of periodic orbits of a given type is minimised by the efficient representatives. To cite this article: Ll. Alsedà et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 479–482.     

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle More concretely, we prove that if , then the period function of the center is monotonically increasing.

     

Monodromy and Stability of a Class of Degenerate Planar Critical Points

Necessary and also sufficient monodromy conditions for a large class of degenerate singular points of planar differential systems are given. For these systems, we also find a computable expression of the principal term of the asymptotic expansion of the return map, which gives the stability of the point.     

Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+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)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.     

Bounding the number of zeros of certain Abelian integrals

In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n+k−1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way.     

On Coxeter recurrences

An interesting family of recurrences of order n ≥ 2, which are globally (n+3)-periodic was introduced by Coxeter in 1971. We prove a surprising property of this family: ‘all’ the possible geometrical behaviours that linear real (n+3)-periodic recurrences can have are present inside the Coxeter recurrences.     

Criteria to bound the number of critical periods

In the present paper we study the period function of centers of potential systems. We obtain criteria to bound the number of critical periods. In case that the system is polynomial, our result enables to tackle the problem from a purely algebraic point of view, since it allows to bound the number of critical periods by counting the zeros of a polynomial. To illustrate its applicability some new and old results are proved.     

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most \(n\geqslant 0\) critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family \(\ddot{x}=x^p-x^q,\) \(p,q\in {\mathbb {R}}\) with \(p>q\).