Space Manifold dynamics

The term Space Manifold Dynamics (SMD) has been proposed for encompassing the various applications of Dynamical Systems methods to spacecraft mission analysis and design, ranging from the exploitation of libration orbits around the collinear Lagrangian points to the design of optimal station-keeping and eclipse avoidance manoeuvres or the determination of low energy lunar and interplanetary transfers

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

Chaotic dynamics and team effectiveness: evidence from professional basketball

Considering teams as complex adaptive systems (CAS) this study deals with changes in team effectiveness over time in a specific context: professional basketball. The sample comprised 23 basketball teams whose outcomes were analysed over a 12-year period according to two objective measures. The results reveal that all the teams showed chaotic dynamics, one of the key characteristics of CAS. A relationship was also found between teams showing low-dimensional chaotic dynamics and better outcomes, supporting the idea of healthy variability in organizational behaviour. The stability of the squad was likewise found to influence team outcomes, although it was not associated with the chaotic dynamics in team effectiveness. It is concluded that studying teams as CAS enables fluctuations in team effectiveness to be explained, and that the techniques derived from nonlinear dynamical systems, developed specifically for the study of CAS, are useful for this purpose.

09014-010 Lytle Creek Watershed Final Report

Brief Project Summary (no greater than this space allows): Leisure Lake is a 20 acre water body located in northwest Jackson County with a 2,581 acre drainage area. This portion of the Maquoketa Watershed including the lake is a tributary to Lytle Creek which drains into the North Fork Maquoketa River and into the Maquoketa Watershed. Portions of the Lytle Creek and North Fork Maquoketa River are on the 303(d) impaired waterbodies list. The project area includes a community of 370 residential properties and one business that currently has no central waste water collection and treatment system. The County Sanitarian estimates at least 225 of these properties do not have properly operating septic systems and ultimately drain their wastewater into the lake. The purpose of this project is to construct a wastewater collection and treatment facility to improve water quality in the creek and river. The project will eliminate the non-permitted septic systems and construct a new wastewater system to properly treat wastewater prior to its discharge into the waterways.

Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model.

Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh¿Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors

Implicit methods for qualitative modeling of gene regulatory networks.

Advancements in high-throughput technologies to measure increasingly complex biological phenomena at the genomic level are rapidly changing the face of biological research from the single-gene single-protein experimental approach to studying the behavior of a gene in the context of the entire genome (and proteome). This shift in research methodologies has resulted in a new field of network biology that deals with modeling cellular behavior in terms of network structures such as signaling pathways and gene regulatory networks. In these networks, different biological entities such as genes, proteins, and metabolites interact with each other, giving rise to a dynamical system. Even though there exists a mature field of dynamical systems theory to model such network structures, some technical challenges are unique to biology such as the inability to measure precise kinetic information on gene-gene or gene-protein interactions and the need to model increasingly large networks comprising thousands of nodes. These challenges have renewed interest in developing new computational techniques for modeling complex biological systems. This chapter presents a modeling framework based on Boolean algebra and finite-state machines that are reminiscent of the approach used for digital circuit synthesis and simulation in the field of very-large-scale integration (VLSI). The proposed formalism enables a common mathematical framework to develop computational techniques for modeling different aspects of the regulatory networks such as steady-state behavior, stochasticity, and gene perturbation experiments.

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}

Weierstrass integrability of differential equations

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.

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.

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.

Suorityskyvyn mittaaminen johtamisen apuvälineenä prosessiorganisaatiossa.

Prosessiorganisaatiossa johtamisen ja prosessien tavoitteena on tyydyttää asiakkaan (sisäinen tai ulkoinen) tarpeet. Mittaamisen sitominen prosessin suorityskyvyn mittaamiseen antaa johdolle kuvan yrityksen toiminnasta. Suorityskykymittariston ja yksittäisten mittareiden avulla yritysjohto pystyy arvioimaan toiminnan tasoa, asettamaan tavoitteita sekä seuraamaan asettamiensa tavoitteiden toteutumista. Työn ensimmäisenä tavoitteena oli kartoittaa edellytyksiä sekä tukea Balanced Scorecardin mukaisen suorityskykymittariston tulevaisuuden implementointia. Mittaristo on tarkoitettu toimitusketjun prosessien tehokkuuden mittaamiseen. Työn toisena tavoitteena oli prosessipohjaisen ajattelutavan tukeminen suorituskykymittariston avulla. Implementoinnin edellytyksiä testattiin valitsemalla kaksi ensimmäisen tason avainmittaria pilottimittareiksi. Varaston suorityskykyä mittaavien pilottimittareiden avulla selvitettiin SC tuoteryhmän osalta toimitusketjun suorituskyky avainasiakkaiden ja tärkeiden markkina-alueiden osalta. Erona käytössä oleviin mittareihin on se, että uudet avainmittarit kattavat koko yrityksen toimitusketjun, kun tällä hetkellä käytössä olevat mittarit mittaavat toimitusketjun yksittäisiä osia. Uusien avaimittareiden lähtöarvot selvitettiin tietokantakyselyjen avulla. Tietokyselyt suoritettiin useissa yksittäisissä tietojärjejestelmissä, jonka jälkeen niiden tulokset koottiin yhteen tiedostoon ja analysoitiin PC sovellusten avulla. Mittauskohteet oli valittu yhdessä linjaorganisaation kanssa. Näin taattiin yhtiön johdon sitoutuminen mittariston kehittämiseen ja käyttöönottoon. Organisaatiossa yksittäisten prosessien (esim. mittaamisprosessi) vastuualueiden selventämiseen käytettiin koeluonteisesti vastuumatriisitekniikkaa. Prosessiorganisaatiossa johtamisen ja prosessien tavoitteena on tyydyttää asiakkaan (sisäinen tai ulkoinen) tarpeet. Mittaamisen sitominen prosessin suorityskyvyn mittaamiseen antaa johdolle kuvan yrityksen toiminnasta. Suorityskykymittariston ja yksittäisten mittareiden avulla yritysjohto pystyy arvioimaan toiminnan tasoa, asettamaan tavoitteita sekä seuraamaan asettamiensa tavoitteiden toteutumista. Työn ensimmäisenä tavoitteena oli kartoittaa edellytyksiä sekä tukea Balanced Scorecardin mukaisen suorityskykymittariston tulevaisuuden implementointia. Mittaristo on tarkoitettu toimitusketjun prosessien tehokkuuden mittaamiseen. Työn toisena tavoitteena oli prosessipohjaisen ajattelutavan tukeminen suorituskykymittariston avulla.Implementoinnin edellytyksiä testattiin valitsemalla kaksi ensimmäisen tason avainmittaria pilottimittareiksi. Varaston suorityskykyä mittaavien pilottimittareiden avulla selvitettiin SC tuoteryhmän osalta toimitusketjun suorituskyky avainasiakkaiden ja tärkeiden markkina-alueiden osalta. Erona käytössä oleviin mittareihin on se, että uudet avainmittarit kattavat koko yrityksen toimitusketjun, kun tällä hetkellä käytössä olevat mittarit mittaavat toimitusketjun yksittäisiä osia. Uusien avaimittareiden lähtöarvot selvitettiin tietokantakyselyjen avulla. Tietokyselyt suoritettiin useissa yksittäisissä tietojärjejestelmissä, jonka jälkeen niiden tulokset koottiin yhteen tiedostoon ja analysoitiin PC sovellusten avulla. Mittauskohteet oli valittu yhdessä linjaorganisaation kanssa. Näin taattiin yhtiön johdon sitoutuminen mittariston kehittämiseen ja käyttöönottoon. Organisaatiossa yksittäisten prosessien (esim. mittaamisprosessi) vastuualueiden selventämiseen käytettiin koeluonteisesti vastuumatriisitekniikkaa.

Dynamic Criteria: a Longitudinal Analysis of Professional Basketball Players" Outcomes

This paper describes the fluctuations of temporal criteria dynamics in the context of professional sport. Specifically, we try to verify the underlying deterministic patterns in the outcomes of professional basketball players. We use a longitudinal approach based on the analysis of the outcomes of 94 basketball players over ten years, covering practically players" entire career development. Time series were analyzed with techniques derived from nonlinear dynamical systems theory. These techniques analyze the underlying patterns in outcomes without previous shape assumptions (linear or nonlinear). These techniques are capable of detecting an intermediate situation between randomness and determinism, called chaos. So they are very useful for the study of dynamic criteria in organizations. We have found most players (88.30%) have a deterministic pattern in their outcomes, and most cases are chaotic (81.92%). Players with chaotic patterns have higher outcomes than players with linear patterns. Moreover, players with power forward and center positions achieve better results than other players. The high number of chaotic patterns found suggests caution when appraising individual outcomes, when coaches try to find the appropriate combination of players to design a competitive team, and other personnel decisions. Management efforts must be made to assume this uncertainty.

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.