SYNCHRONIZATION OF CHAOS AND THE TRANSITION TO WAVE TURBULENCE

We investigated the transition to wave turbulence in a spatially extended three-wave interacting model, where a spatially homogeneous state undergoing chaotic dynamics undergoes spatial mode excitation. The transition to this weakly turbulent state can be regarded as the loss of synchronization of chaos of mode oscillators describing the spatial dynamics.

ACTIVE NETWORKS THAT MAXIMIZE THE AMOUNT OF INFORMATION TRANSMISSION

This work clarifies the relationship between network circuit (topology) and behavior (information transmission and synchronization) in active networks, e. g. neural networks. As an application, we show how to determine a network topology that is optimal for information transmission. By optimal, we mean that the network is able to transmit a large amount of information, it possesses a large number of communication channels, and it is robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.

Regularity at infinity of real mappings and a Morse-Sard theorem

We prove a new Morse-Sard-type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for C-2 mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role of our picture is played by the p-regularity and its bridge toward the rho-regularity which implies topological triviality at infinity.

STRUCTURE AND DYNAMICS: THE TRANSITION FROM NONEQUILIBRIUM TO EQUILIBRIUM IN INTEGRATE-AND-FIRE DYNAMICS

The transient and equilibrium properties of dynamics unfolding in complex systems can depend critically on specific topological features of the underlying interconnections. In this work, we investigate such a relationship with respect to the integrate-and-fire dynamics emanating from a source node and an extended network model that allows control of the small-world feature as well as the length of the long-range connections. A systematic approach to investigate the local and global correlations between structural and dynamical features of the networks was adopted that involved extensive simulations (one and a half million cases) so as to obtain two-dimensional correlation maps. Smooth, but diverse surfaces of correlation values were obtained in all cases. Regarding the global cases, it has been verified that the onset avalanche time (but not its intensity) can be accurately predicted from the structural features within specific regions of the map (i.e. networks with specific structural properties). The analysis at local level revealed that the dynamical features before the avalanches can also be accurately predicted from structural features. This is not possible for the dynamical features after the avalanches take place. This is so because the overall topology of the network predominates over the local topology around the source at the stationary state.

TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Modeling and identification of freeplay nonlinearity

A nonlinear analysis is performed for the purpose of identification of the pitch freeplay nonlinearity and its effect on the type of bifurcation of a two degree-of-freedom aeroelastic system. The databases for the identification are generated from experimental investigations of a pitch-plunge rigid airfoil supported by a nonlinear torsional spring. Experimental data and linear analysis are performed to validate the parameters of the linearized equations. Based on the periodic responses of the experimental data which included the flutter frequency and its third harmonics, the freeplay nonlinearity is approximated by a polynomial expansion up to the third order. This representation allows us to use the normal form of the Hopf bifurcation to characterize the type of instability. Based on numerical integrations, the coefficients of the polynomial expansion representing the freeplay nonlinearity are identified. Published by Elsevier Ltd.

Dynamical changes from harmonic vibrations of a limited power supply driving a Duffing oscillator

The harmonic oscillations of a Duffing oscillator driven by a limited power supply are investigated as a function of the alternative strength of the rotor. The semi-trivial and non-trivial solutions are derived. We examine the stability of these solutions and then explore the complex behaviors associated with the bifurcations sequences. Interestingly, a 3D diagram provides a global view of the effects of alternate strength on the appearance of chaos and hyperchaos on the system.

An anatomical analysis of the mini-modified orbitozygomatic and supra-orbital approaches

Seven sides of cadaver heads were used to compare the surgical exposures provided by the mini-modified orbitozygomatic (MOz) and supra-orbital (SO) approaches. The Optotrak 3020 computerized tracking system (Northern Digital, Waterloo, ON, Canada) was utilized to evaluate the area of anatomical exposure defined by six points: (1) ipsilateral sphenoid ridge; (2) most distal point of the ipsilateral middle cerebral artery (MCA); (3) most distal point of the ipsilateral posterior cerebral artery (PCA); (4) most distal point of the contralateral PCA; (5) most distal point of the contralateral MCA; and (6) contralateral sphenoid ridge. Additionally, angles of approach for the ipsilateral MCA bifurcation, ipsilateral ICA bifurcation, basilar artery tip, contralateral MCA and ICA bifurcation and anterior communicating artery (AcomA) were evaluated, first for SO and then for MOz. An image guidance system was used to evaluate the limits of surgical exposure. No differences in the area of surgical exposure were noted (p > 0.05). Vertical angles were significantly wider for the ipsilateral and contralateral ICA bifurcation, AcomA, contralateral MCA and basilar tip (p < 0.05) for MOz. No differences in horizontal angles were observed between the approaches for the six targets (p > 0.05). There were no differences in the limits of exposure. MOz affords no additional surgical working space. However, our results demonstrate systematically that vertical exposure is improved. The MOz should be performed while planning an approach to these regions and a wider exposure in the vertical axis is needed. (C) 2012 Elsevier Ltd. All rights reserved.

Analysis of nonlinear dynamics of vocal folds using high-speed video observation and biomechanical modeling

Primary voice production occurs in the larynx through vibrational movements carried out by vocal folds. However, many problems can affect this complex system resulting in voice disorders. In this context, time-frequency-shape analysis based on embedding phase space plots and nonlinear dynamics methods have been used to evaluate the vocal fold dynamics during phonation. For this purpose, the present work used high-speed video to record the vocal fold movements of three subjects and extract the glottal area time series using an image segmentation algorithm. This signal is used for an optimization method which combines genetic algorithms and a quasi-Newton method to optimize the parameters of a biomechanical model of vocal folds based on lumped elements (masses, springs and dampers). After optimization, this model is capable of simulating the dynamics of recorded vocal folds and their glottal pulse. Bifurcation diagrams and phase space analysis were used to evaluate the behavior of this deterministic system in different circumstances. The results showed that this methodology can be used to extract some physiological parameters of vocal folds and reproduce some complex behaviors of these structures contributing to the scientific and clinical evaluation of voice production. (C) 2010 Elsevier Inc. All rights reserved.

The tangential variation of a localized flux-type eigenvalue problem

In this work the differentiability of the principal eigenvalue lambda = lambda(1)(Gamma) to the localized Steklov problem -Delta u + qu = 0 in Omega, partial derivative u/partial derivative nu = lambda chi(Gamma)(x)u on partial derivative Omega, where Gamma subset of partial derivative Omega is a smooth subdomain of partial derivative Omega and chi(Gamma) is its characteristic function relative to partial derivative Omega, is shown. As a key point, the flux subdomain Gamma is regarded here as the variable with respect to which such differentiation is performed. An explicit formula for the derivative of lambda(1) (Gamma) with respect to Gamma is obtained. The lack of regularity up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem. Therefore, the whole analysis must be done in the weak sense of H(1)(Omega). The study is of interest in mathematical models in morphogenesis. (C) 2011 Elsevier Inc. All rights reserved.

Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis

In order to achieve a better understanding of multiple infections and long latency in the dynamics of Mycobacterium tuberculosis infection, we analyze a simple model. Since backward bifurcation is well documented in the literature with respect to the model we are considering, our aim is to illustrate this behavior in terms of the range of variations of the model's parameters. We show that backward bifurcation disappears (and forward bifurcation occurs) if: (a) the latent period is shortened below a critical value; and (b) the rates of super-infection and re-infection are decreased. This result shows that among immunosuppressed individuals, super-infection and/or changes in the latent period could act to facilitate the onset of tuberculosis. When we decrease the incubation period below the critical value, we obtain the curve of the incidence of tuberculosis following forward bifurcation; however, this curve envelops that obtained from the backward bifurcation diagram.

Algorithms for network design and routing problems

In this thesis we study three combinatorial optimization problems belonging to the classes of Network Design and Vehicle Routing problems that are strongly linked in the context of the design and management of transportation networks: the Non-Bifurcated Capacitated Network Design Problem (NBP), the Period Vehicle Routing Problem (PVRP) and the Pickup and Delivery Problem with Time Windows (PDPTW). These problems are NP-hard and contain as special cases some well known difficult problems such as the Traveling Salesman Problem and the Steiner Tree Problem. Moreover, they model the core structure of many practical problems arising in logistics and telecommunications. The NBP is the problem of designing the optimum network to satisfy a given set of traffic demands. Given a set of nodes, a set of potential links and a set of point-to-point demands called commodities, the objective is to select the links to install and dimension their capacities so that all the demands can be routed between their respective endpoints, and the sum of link fixed costs and commodity routing costs is minimized. The problem is called non- bifurcated because the solution network must allow each demand to follow a single path, i.e., the flow of each demand cannot be splitted. Although this is the case in many real applications, the NBP has received significantly less attention in the literature than other capacitated network design problems that allow bifurcation. We describe an exact algorithm for the NBP that is based on solving by an integer programming solver a formulation of the problem strengthened by simple valid inequalities and four new heuristic algorithms. One of these heuristics is an adaptive memory metaheuristic, based on partial enumeration, that could be applied to a wider class of structured combinatorial optimization problems. In the PVRP a fleet of vehicles of identical capacity must be used to service a set of customers over a planning period of several days. Each customer specifies a service frequency, a set of allowable day-combinations and a quantity of product that the customer must receive every time he is visited. For example, a customer may require to be visited twice during a 5-day period imposing that these visits take place on Monday-Thursday or Monday-Friday or Tuesday-Friday. The problem consists in simultaneously assigning a day- combination to each customer and in designing the vehicle routes for each day so that each customer is visited the required number of times, the number of routes on each day does not exceed the number of vehicles available, and the total cost of the routes over the period is minimized. We also consider a tactical variant of this problem, called Tactical Planning Vehicle Routing Problem, where customers require to be visited on a specific day of the period but a penalty cost, called service cost, can be paid to postpone the visit to a later day than that required. At our knowledge all the algorithms proposed in the literature for the PVRP are heuristics. In this thesis we present for the first time an exact algorithm for the PVRP that is based on different relaxations of a set partitioning-like formulation. The effectiveness of the proposed algorithm is tested on a set of instances from the literature and on a new set of instances. Finally, the PDPTW is to service a set of transportation requests using a fleet of identical vehicles of limited capacity located at a central depot. Each request specifies a pickup location and a delivery location and requires that a given quantity of load is transported from the pickup location to the delivery location. Moreover, each location can be visited only within an associated time window. Each vehicle can perform at most one route and the problem is to satisfy all the requests using the available vehicles so that each request is serviced by a single vehicle, the load on each vehicle does not exceed the capacity, and all locations are visited according to their time window. We formulate the PDPTW as a set partitioning-like problem with additional cuts and we propose an exact algorithm based on different relaxations of the mathematical formulation and a branch-and-cut-and-price algorithm. The new algorithm is tested on two classes of problems from the literature and compared with a recent branch-and-cut-and-price algorithm from the literature.

Verzweigung periodischer Lösungen bei rein nichtlinearen Differentialgleichungssystemen in der Ebene

Zusammenfassung:In dieser Arbeit werden die Abzweigung stationärer Punkte und periodischer Lösungen von isolierten stationären Punkten rein nichtlinearer Differentialgleichungen in der reellenEbene betrachtet.Das erste Kapitel enthält einige technische Hilfsmittel, während im zweiten ausführlich das Verhalten von Differentialgleichungen in der Ebene mit zwei homogenen Polynomen gleichen Grades als rechter Seite diskutiert wird.Im dritten Kapitel beginnt der Hauptteil der Arbeit. Hier wird eine Verallgemeinerung des Hopf'schen Verzweigungssatzes bewiesen, der den klassischen Satz als Spezialfall enthält.Im vierten Kapitel untersuchen wir die Abzweigung stationärer Punkte und im letzten Kapitel die Abzweigung periodischer Lösungen unter Störungen, deren Ordnung echt kleiner ist, als die erste nichtverschwindende Näherung der ungestörten Gleichung.Alle Voraussetzungen in dieser Arbeit sind leicht nachzurechnen und es werden zahlreiche Beispiele ausführlich diskutiert.

Parametric Sensitivity Analysis of the Most Recent Computational Models of Rabbit Cardiac Pacemaking

The cellular basis of cardiac pacemaking activity, and specifically the quantitative contributions of particular mechanisms, is still debated. Reliable computational models of sinoatrial nodal (SAN) cells may provide mechanistic insights, but competing models are built from different data sets and with different underlying assumptions. To understand quantitative differences between alternative models, we performed thorough parameter sensitivity analyses of the SAN models of Maltsev & Lakatta (2009) and Severi et al (2012). Model parameters were randomized to generate a population of cell models with different properties, simulations performed with each set of random parameters generated 14 quantitative outputs that characterized cellular activity, and regression methods were used to analyze the population behavior. Clear differences between the two models were observed at every step of the analysis. Specifically: (1) SR Ca2+ pump activity had a greater effect on SAN cell cycle length (CL) in the Maltsev model; (2) conversely, parameters describing the funny current (If) had a greater effect on CL in the Severi model; (3) changes in rapid delayed rectifier conductance (GKr) had opposite effects on action potential amplitude in the two models; (4) within the population, a greater percentage of model cells failed to exhibit action potentials in the Maltsev model (27%) compared with the Severi model (7%), implying greater robustness in the latter; (5) confirming this initial impression, bifurcation analyses indicated that smaller relative changes in GKr or Na+-K+ pump activity led to failed action potentials in the Maltsev model. Overall, the results suggest experimental tests that can distinguish between models and alternative hypotheses, and the analysis offers strategies for developing anti-arrhythmic pharmaceuticals by predicting their effect on the pacemaking activity.

The effects of hypocalcemia on spatial alternans and ventricular fibrillation studied with the optical mapping technique

The heart is a wonderful but complex organ: it uses electrochemical mechanisms in order to produce mechanical energy to pump the blood throughout the body and allow the life of humans and animals. This organ can be subject to several diseases and sudden cardiac death (SCD) is the most catastrophic manifestation of these diseases, responsible for the death of a large number of people throughout the world. It is estimated that 325000 Americans annually die for SCD. SCD most commonly occurs as a result of reentrant tachyarrhythmias (ventricular tachycardia (VT) and ventricular fibrillation (VF)) and the identification of those patients at higher risk for the development of SCD has been a difficult clinical challenge. Nowadays, a particular electrocardiogram (ECG) abnormality, “T-wave alternans” (TWA), is considered a precursor of lethal cardiac arrhythmias and sudden death, a sensitive indicator of risk for SCD. TWA is defined as a beat-to-beat alternation in the shape, amplitude, or timing of the T-wave on the ECG, indicative of the underlying repolarization of cardiac cells [5]. In other words TWA is the macroscopic effect of subcellular and celluar mechanisms involving ionic kinetics and the consequent depolarization and repolarization of the myocytes. Experimental activities have shown that TWA on the ECG is a manifestation of an underlying alternation of long and short action potential durations (APDs), the so called APD-alternans, of cardiac myocytes in the myocardium. Understanding the mechanism of APDs-alternans is the first step for preventing them to occur. In order to investigate these mechanisms it’s very important to understand that the biological systems are complex systems and their macroscopic properties arise from the nonlinear interactions among the parts. The whole is greater than the sum of the parts, and it cannot be understood only by studying the single parts. In this sense the heart is a complex nonlinear system and its way of working follows nonlinear dynamics; alternans also, they are a manifestation of a phenomenon typical in nonlinear dynamical systems, called “period-dubling bifurcation”. Over the past decade, it has been demonstrated that electrical alternans in cardiac tissue is an important marker for the development of ventricular fibrillation and a significant predictor for mortality. It has been observed that acute exposure to low concentration of calcium does not decrease the magnitude of alternans and sustained ventricular Fibrillation (VF) is still easily induced under these condition. However with prolonged exposure to low concentration of calcium, alternans disappears, but VF is still inducible. This work is based on this observation and tries to make it clearer. The aim of this thesis is investigate the effect of hypocalcemia spatial alternans and VF doing experiments with canine hearts and perfusing them with a solution with physiological ionic concentration and with a solution with low calcium concentration (hypocalcemia); in order to investigate the so called memory effect, the experimental activity was modified during the way. The experiments were performed with the optical mapping technique, using voltage-sensitive dye, and a custom made Java code was used in post-processing. Finding the Nolasco and Dahlen’s criterion [8] inadequate for the prediction of alternans, and takin into account the experimental results, another criterion, which consider the memory effect, has been implemented. The implementation of this criterion could be the first step in the creation of a method, AP-based, discriminating who is at risk if developing VF. This work is divided into four chapters: the first is a brief presentation of the physiology of the heart; the second is a review of the major theories and discovers in the study of cardiac dynamics; the third chapter presents an overview on the experimental activity and the optical mapping technique; the forth chapter contains the presentation of the results and the conclusions.