Periodic orbits of a one-dimensional non-autonomous Hamiltonian system

In this paper we study the properties of the periodic orbits of with x∈S¹ and a T₀ periodic potential. Called the frequency of windings of an orbit in S¹ we show that exists an infinite number of periodic solutions with a given ρ. We give a lower bound on the number of periodic orbits with a given period and ρ by means of the Morse theory.     

Continuous orbit transitions in a one-dimensional inelastic particle system

Continuous transitions between different periodic orbits in a one-dimensional inelastic particle system with two particles are investigated. We explain why continuous transitions that occur when adding or subtracting a single collision are, generically, of co-dimension 2. However, we show that there are an infinite set of degenerate transitions of co-dimension 1. We provide an analysis that gives a simple criteria to classify which transitions are degenerated purely from the discrete set of collisions that occur in the orbits.     

Periodic orbits and expansiveness

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles.     

Periodic orbits from nonperiodic orbits on an interval

This paper demonstrates that any continuous real-valued function which has an orbit with infinitely many limit points must necessarily have periodic cycles of arbitrarily large prime period. We present an example of a function with an orbit whose limit points are exactly Z⁺.     

Periodic orbits in the case of a zero eigenvalue

We will show that if a dynamical system has enough constants of motion then a Moser–Weinstein type theorem can be applied for proving the existence of periodic orbits in the case when the linearized system is degenerate. To cite this article: P. Birtea et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

Periodic orbits for a class ofC 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in\(\mathbb{R}^3 \). We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in\(\mathbb{R}^3 \) with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied.     

Periodic orbits for additive cellular automata

We formulate and study a necessary and sufficient condition for a configuration of any type of infinite additive cellular automata to have periodic behavior in time. The number of orbits with periodn is counted. Relations between spatial and temporal periods are discussed.Supported in part by G.M.C.I., DEEE-LNETI (Portugal).     

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.     

Periodic orbits of competitive and cooperative systems

In this paper we consider the existence, location and stability type of periodic orbits of competitive and cooperative systems of autonomous ordinary differential equations. Particular attention is given to the existence of invariant manifolds related to periodic orbits and these results are used to improve a result of Hirsch for three dimensional irreducible competitive and cooperative systems. In particular, the Poincaré-Bendixson theorem holds for such three dimensional systems.     

Periodic orbits of the restricted three-body problem

We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period , for every . We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.

     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Periodic orbits of delay differential equations under discretization

This paper deals with the long-time behaviour of numerical solutions of delay differential equations that have asymptotically stable periodic orbits. It is shown that Runge-Kutta discretizations of such equations have attractive invariant curves which approximate the periodic orbit with the order of the method. The research by this author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

Periodic monotone systems having one-dimensional dynamics

We establish a criterion for a periodic monotone system to display a fully one dimensional dynamical behavior. This criterion is based on the existence of a Lyapunov function acting on differences of unordered solutions. The main consequence is the convergence of every solution to a periodic one.