Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions

AbstractIn 1964, Sarkovskii defined a certain linear ordering ⩽s of the positive integers and proved that m⩽sn if every continuous f:R→R having an orbit of size n also has an orbit of size m. This idea is extended to get a partial (but not linear) ordering in which the pattern of the orbit is taken into account. For example if x1<x2<x3<x4, then x1→x2→x3→x4x→1 and x1→x3→x2→x4→x1 are both orbits of size 4 but are considered to have distinct patterns in this paper. A combinatorial algorithm which decides the status of any two patterns with respect to the partial ordering is derived, and examples are given for patterns of small size

Adiabatic dynamical systems and hysteresis

This work is dedicated to the study of Dynamical Systems, depending on a slowly varying parameter. It contains, in particular, a detailed analysis of memory effects, such as hysteresis, which frequently appear in systems involving several time scales. In a first part of this dissertation, we develop a mathematical framework to deal with adiabatic differential equations. We do this, whenever possible, by favouring the geometrical approach to the theory, which allows to derive qualitative properties of the dynamics, such as existence of hysteresis cycles and scaling laws, with a minimum of analytic calculations. We begin by analysing one-dimensional adiabatic systems of the form ε&#7819; = f(x,τ). We first show existence of adiabatic solutions, which remain close to equilibrium branches of the system, and admit asymptotic series in the adiabatic parameter ε. We then provide a method to analyse solutions near bifurcation points, and show that they scale in a nontrivial way with ε, with an exponent that can be easily computed. The analysis is concluded by examining global properties of the flow, in particular existence of hysteresis cycles. These results are then extended to the n-dimensional case. The discussion of adiabatic solutions carries over in a natural way. The dynamics of neighboring solutions is, however, more difficult to analyse. We first provide a method to diagonalize linear equations dynamically, and show that eigenvalue crossings lead to similar behaviours than bifurcations. We then introduce some methods to deal with nonlinear terms, in particular adiabatic manifolds and dynamic normal forms. In a second part of this work, we apply the previously developed methods to some selected examples. We first discuss the dynamics of some low-dimensional nonlinear oscillators. In particular, we present the example of a damped pendulum, on a table rotating with a slowly oscillating angular frequency. This system displays chaotic motion even for arbitrarily small adiabatic parameter. This phenomenon is explained by computing an asymptotic expression of the Poincaré map. As a second application, we analyse a few models of ferromagnetism. Starting from a lattice model with stochastic spin flip dynamics, we show how to derive a deterministic equation of motion of Ginzburg-Landau type, in the case of infinite range interactions and in the thermodynamic limit. We analyse the influence of dimensionality and interaction anisotropy on shape and scaling properties of hysteresis cycles. A few simple approximations to the dynamics of an Ising model are also discussed. We conclude this work by extending some properties of adiabatic differential equations to iterated maps. We give some results on existence of adiabatic invariants for near-integrable slow-fast maps, and apply them to billiards

On the minimum positive entropy for cycles on trees

Consider, for any n ∈ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn ( Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial xn − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = mk, where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn

On the minimum positive entropy for cycles on trees

Consider, for any n ∈ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn ( Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial xn − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = mk, where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn