Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences

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.

Algebraic decay of velocity fluctuations near a wall

Computer simulations of the dynamics of a colloidal particle suspended in a fluid confined by an interface show that the asymptotic decay of the velocity correlation functions is algebraic. The exponents of the long-time tails depend on the direction of motion of the particle relative to the surface, as well as on the specific nature of the boundary conditions. In particular, we find that for the angular velocity correlation function, the decay in the presence of a slip surface is faster than the one corresponding to a stick one. An intuitive picture is introduced to explain the various long-time tails, and the simulations are compared with theoretical expressions where available.

Notas del primer seminario de integrabilidad de la Universitat Politècnica de Catalunya

Estas notas corresponden a las exposiciones presentadas en el \emph{Primer Seminario de Integrabilidad}, dentro de lo que se denomina \emph{Aula de Sistemas Din\'amicos}. Durante este evento se realizaron seis conferencias, todas presentadas por miembros del grupo de Sistemas Din\'amicos de la UPC. El programa desarrollado fue el siguiente:\\\begin{center}AULA DE SISTEMAS DIN\'AMICOS\end{center}\begin{center}\texttt{http://www.ma1.upc.es/recerca/seminaris/aulasd-cat.html}\end{center}\begin{center}SEMINARIO DE INTEGRABILIDAD\end{center}\begin{center}Martes 29 y Mi\'ercoles 30 de marzo de 2005\\Facultad de Matem\'aticas y Estad\'{\i}stica, UPC\\Aula: Seminario 1\end{center}\bigskip\begin{center}PROGRAMA Y RES\'UMENES\end{center}{\bf Martes 29 de marzo}\begin{itemize}\item15:30. Juan J. Morales-Ruiz. \emph{El problema de laintegrabilidad en Sistemas Din\'amicos}\medskip {\bf Resumen.} En esta presentaci\'on se pretende dar unaidea de conjunto, pero sin entrar en detalles, sobre las diversasnociones de integrabilidad, asociadas a nombres de matem\'aticostan ilustres como Liouville, Galois-Picard-Vessiot, Lie, Darboux,Kowalevskaya, Painlev\'e, Poincar\'e, Kolchin, Lax, etc. Adem\'astambi\'en mencionaremos la revoluci\'on que supuso en los a\~nossesenta del siglo pasado el descubrimiento de Gardner, Green,Kruskal y Miura sobre un nuevo m\'etodo para resolver en algunoscasos determinadas ecuaciones en derivadas parciales. \medskip\item16:00. David G\'omez-Ullate. \emph{Superintegrabilidad, pares deLax y modelos de $N-$cuerpos en el plano}\medskip{\bf Resumen.} Introduciremos algunas t\'ecnicas cl\'asicas paraconstruir modelos de N-cuerpos integrables, como los pares de Laxo la din\'amica de los ceros de un polinomio. Revisaremos lanoci\'on de integrabilidad Liouville y superintegrabilidad, ydiscutiremos un nuevo m\'etodo debido a F. Calogero para contruirmodelos de N-cuerpos en el plano con muchas \'orbitasperi\'odicas. La exposici\'on se acompa\~nar\'a de animaciones delmovimiento de los cuerpos, y se plantear\'an algunos problemasabiertos.\medskip\item17:00. Pausa\medskip\item17:30. Yuri Fedorov. \emph{An\'alisis de Kovalevskaya--Painlev\'ey Sistemas Algebraicamente Integrables}\medskip{\bf Resumen.} Muchos sistemas integrables poseen una propiedadremarcable: todas sus soluciones son funciones meromorfas deltiempo como una variable compleja. Tal comportamiento, que serefiere como propiedad de Kovalevskaya-Painleve (KP) y que se usafrecuentemente como una ensayo de integrabilidad, no es accidentaly tiene unas ra\'{\i}ces geom\'etricas profundas. En esta charladescribiremos una clase de tales sistemas (conocidos como lossistemas algebraicamente integrables) y subrayaremos suspropiedades geom\'etricas principales que permiten predecir laestructura de las soluciones complejas y adem\'as encontrarlasexpl\'{\i}citamente. Eso lo ilustraremos con algunos sistemas dela mec\'anica cl\'asica. Tambi\'en mencionaremos unasgeneralizaciones \'utiles de la noci\'on de integrabilidadalgebraica y de la propiedad KP.\end{itemize}\medskip{\bf Mi\'ercoles 30 de marzo}\begin{itemize}\item 15:30. Rafael Ram\'{\i}rez-Ros. \emph{El m\'etodo de Poincar\'e}\medskip{\bf Resumen.} Dado un sistema Hamiltoniano aut\'onomo cercano acompletamente integrable Poincar\'e prob\'o que, en general, noexiste ninguna integral primera adicional uniforme en elpar\'ametro de perturbaci\'on salvo el propio Hamiltoniano.Esbozaremos las ideas principales del m\'etodo de prueba ycomentaremos algunas extensiones y generalizaciones.\newpage\item16:30. Chara Pantazi. \emph{El M\'etodo de Darboux}\medskip{\bf Resumen.} Darboux, en 1878, present\'o su m\'etodo paraconstruir integrales primeras de campos vectoriales polinomialesutilizando sus curvas invariantes algebraicas. En estaexposici\'on presentaremos algunas extensiones del m\'etodocl\'asico de Darboux y tambi\'en algunas aplicaciones.\medskip\item17:30. Pausa\medskip\item18:00. Juan J. Morales-Ruiz. \emph{M\'etodos recientes paradetectar la no integrabilidad}\medskip{\bf Resumen.} En 1982 Ziglin utiliza la estructura de laecuaci\'on en variaciones de Poincar\'e (sobre una curva integralparticular) como una herramienta fundamental para detectar la nointegrabilidad de un sistema Hamiltoniano. En esta charla sepretende dar una idea de esta aproximaci\'on a la nointegrabilidad, junto con t\'ecnicas m\'as recientes queinvolucran la teor\'{\i}a de Galois de ecuaciones diferencialeslineales, haciendo \'enfasis en los ejemplos m\'as que en lateor\'{\i}a general. Ilustraremos estos m\'etodos con resultadossobre la no integrabilidad de algunos problemas de $N$ cuerpos enMec\'anica Celeste.\end{itemize}

Cyclicity versus center problem

We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate this phenomenon in several examples for which this cyclicity is computed.

A survey on the inverse integrating factor

The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relationwith limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.

On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

Persistent random walk model for transport trough thin slabs

We present a model for transport in multiply scattering media based on a three-dimensional generalization of the persistent random walk. The model assumes that photons move along directions that are parallel to the axes. Although this hypothesis is not realistic, it allows us to solve exactly the problem of multiple scattering propagation in a thin slab. Among other quantities, the transmission probability and the mean transmission time can be calculated exactly. Besides being completely solvable, the model could be used as a benchmark for approximation schemes to multiple light scattering.

A decision-making Fokker-Planck model in computational neuroscience

Minimal models for the explanation of decision-making in computational neuroscience are based on the analysis of the evolution for the average firing rates of two interacting neuron populations. While these models typically lead to multi-stable scenario for the basic derived dynamical systems, noise is an important feature of the model taking into account finite-size effects and robustness of the decisions. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker-Planck partial differential equation. In particular, we discuss the existence, positivity and uniqueness for the solution of the stationary equation, as well as for the time evolving problem. Moreover, we prove convergence of the solution to the the stationary state representing the probability distribution of finding the neuron families in each of the decision states characterized by their average firing rates. Finally, we propose a numerical scheme allowing for simulations performed on the Fokker-Planck equation which are in agreement with those obtained recently by a moment method applied to the stochastic differential system. Our approach leads to a more detailed analytical and numerical study of this decision-making model in computational neuroscience.

Necessary and sufficient conditions for robust stability using modal intervals

In this paper, robustness of parametric systems is analyzed using a new approach to interval mathematics called Modal Interval Analysis. Modal Intervals are an interval extension that, instead of classic intervals, recovers some of the properties required by a numerical system. Modal Interval Analysis not only simplifies the computation of interval functions but allows semantic interpretation of their results. Necessary, sufficient and, in some cases, necessary and sufficient conditions for robust performance are presented

An Interval Intelligent-based Approach for Fault Detection and Modelling

Not considered in the analytical model of the plant, uncertainties always dramatically decrease the performance of the fault detection task in the practice. To cope better with this prevalent problem, in this paper we develop a methodology using Modal Interval Analysis which takes into account those uncertainties in the plant model. A fault detection method is developed based on this model which is quite robust to uncertainty and results in no false alarm. As soon as a fault is detected, an ANFIS model is trained in online to capture the major behavior of the occurred fault which can be used for fault accommodation. The simulation results understandably demonstrate the capability of the proposed method for accomplishing both tasks appropriately

Dynamic diagnosis based on interval analytical redundancy relations and signs of the symptoms

A model-based approach for fault diagnosis is proposed, where the fault detection is based on checking the consistencyof the Analytical Redundancy Relations (ARRs) using an interval tool. The tool takes into account the uncertainty in theparameters and the measurements using intervals. Faults are explicitly included in the model, which allows for the exploitation of additional information. This information is obtained from partial derivatives computed from the ARRs. The signs in the residuals are used to prune the candidate space when performing the fault diagnosis task. The method is illustrated using a two-tank example, in which these aspects are shown to have an impact on the diagnosis and fault discrimination, since the proposed method goes beyond the structural methods

Robust fault detection using consistency techniques for uncertainty handling

Often practical performance of analytical redundancy for fault detection and diagnosis is decreased by uncertainties prevailing not only in the system model, but also in the measurements. In this paper, the problem of fault detection is stated as a constraint satisfaction problem over continuous domains with a big number of variables and constraints. This problem can be solved using modal interval analysis and consistency techniques. Consistency techniques are then shown to be particularly efficient to check the consistency of the analytical redundancy relations (ARRs), dealing with uncertain measurements and parameters. Through the work presented in this paper, it can be observed that consistency techniques can be used to increase the performance of a robust fault detection tool, which is based on interval arithmetic. The proposed method is illustrated using a nonlinear dynamic model of a hydraulic system