Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

This paper studies non-autonomous Lyness type recurrences of the form x_{n+2}=(a_n+x_n)/x_{n+1}, where a_n is a k-periodic sequence of positive numbers with prime period k. We show that for the cases k in {1,2,3,6} the behavior of the sequence x_n is simple(integrable) while for the remaining cases satisfying k not a multiple of 5 this behavior can be much more complicated(chaotic). The cases k multiple of 5 are studied separately.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features.

On periodic solutions of 2-periodic Lyness difference equations

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.

J2 effect and elliptic inclined periodic orbits in the collision three-body problem

The existence of a new class of inclined periodic orbits of the collision restricted three-body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet

Profile Following for Inspection of Underwater Structures

We present a seabed profile estimation and following method for close proximity inspection of 3D underwater structures using autonomous underwater vehicles (AUVs). The presented method is used to determine a path allowing the AUV to pass its sensors over all points of the target structure, which is known as coverage path planning. Our profile following method goes beyond traditional seabed following at a safe altitude and exploits hovering capabilities of recent AUV developments. A range sonar is used to incrementally construct a local probabilistic map representation of the environment and estimates of the local profile are obtained via linear regression. Two behavior-based controllers use these estimates to perform horizontal and vertical profile following. We build upon these tools to address coverage path planning for 3D underwater structures using a (potentially inaccurate) prior map and following cross-section profiles of the target structure. The feasibility of the proposed method is demonstrated using the GIRONA 500 AUV both in simulation using synthetic and real-world bathymetric data and in pool trials

Dissemination of periodic mammography and patterns of use, by birth cohort, in Catalonia (Spain)

Background: In Catalonia (Spain) breast cancer mortality has declined since the beginning of the 1990s. The dissemination of early detection by mammography and the introduction of adjuvant treatments are among the possible causes of this decrease, and both were almost coincident in time. Thus, understanding how these procedures were incorporated into use in the general population and in women diagnosed with breast cancer is very important for assessing their contribution to the reduction in breast cancer mortality. In this work we have modeled the dissemination of periodic mammography and described repeat mammography behavior in Catalonia from 1975 to 2006. Methods: Cross-sectional data from three Catalan Health Surveys for the calendar years 1994, 2002 and 2006 was used. The dissemination of mammography by birth cohort was modeled using a mixed effects model and repeat mammography behavior was described by age and survey year. Results: For women born from 1938 to 1952, mammography clearly had a period effect, meaning that they started to have periodic mammograms at the same calendar years but at different ages. The age at which approximately 50% of the women were receiving periodic mammograms went from 57.8 years of age for women born in 1938–1942 to 37.3 years of age for women born in 1963–1967. Women in all age groups experienced an increase in periodic mammography use over time, although women in the 50–69 age group have experienced the highest increase. Currently, the target population of the Catalan Breast Cancer Screening Program, 50–69 years of age, is the group that self-reports the highest utilization of periodic mammograms, followed by the 40–49 age group. A higher proportion of women of all age groups have annual mammograms rather than biennial or irregular ones. Conclusion: Mammography in Catalonia became more widely implemented during the 1990s. We estimated when cohorts initiated periodic mammograms and how frequently women are receiving them. These two pieces of information will be entered into a cost-effectiveness model of early detection in Catalonia.

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity

In two previous papers [J. Differential Equations, 228 (2006), pp. 530 579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261 1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapunov multipliers which are natural measures of hyperbolicity depend on the parameters, with power laws with universal exponents. We also observe that, even if the rigorous justifications in [J. Differential Equations, 228 (2006), pp. 530-579] are developed only for hyperbolic tori, the algorithms work also for elliptic tori in Hamiltonian systems. We can continue these tori and also compute some bifurcations at resonance which may lead to the existence of hyperbolic tori with nonorientable bundles. We compute manifolds tangent to nonorientable bundles.

Analysis of the process of inspection in recreational craft in Spain. Suggestions for safety improvement

Inspections of pleasure boats in Spain can be carried out by collaborating entities of inspection, entities that must be authorized by the Maritime Administration. This authorization allows to perform effective inspections and technical controls of recreational crafts. Recreational crafts are subjected to surveys that are based on the registration list and on the material used in the hull. In addition, required safety equipment of the recreational boat depends on the distance that the recreational boat is authorized to navigate. Following data obtained from inspections of recreational craft, this paper aims to analyze information about hulls within dry and afloat conditions, about the equipment for rescue and safety, and about other nautical equipment; as well as to perform and improve different verifications during the inspections. All this information points to several aspects relevant for the optimization of the inspection process, the ultimate target being increasing efficiency and effectiveness, and ensuring more safety in recreational craft.

Families of periodic horseshoe orbits in the restricted three-body problem

We compute families of symmetric periodic horseshoe orbits in the restricted three-body problem. Both the planar and three-dimensional cases are considered and several families are found.We describe how these families are organized as well as the behavior along and among the families of parameters such as the Jacobi constant or the eccentricity. We also determine the stability properties of individual orbits along the families. Interestingly, we find stable horseshoe-shaped orbit up to the quite high inclination of 17◦

A Note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits