abstract: In this work, we focused on the stability and reducibility of quasi-periodic systems. We examined the quasi-periodic linear Mathieu equation of the form x ̈+(ä+ϵ[cost+cosùt])x=0 The stability of solutions of Mathieu's equation as a function of parameter values (ä,ϵ) had been analyzed in this work. We used the Floquet type theory to generate stability diagrams which were used to determine the bounded regions of stability in the ä-ù plane for fixed ϵ. In the case of reducibility, we first applied the Lyapunov- Floquet (LF) transformation and modal transformation, which converted the linear part of the system into the Jordan form. Very importantly, quasi-periodic near-identity transformation was applied to reduce the system equations to a constant coefficient system by solving homological equations via harmonic balance. In this process we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system to a constant one. / Dissertation/Thesis / M.S.Tech Engineering 2012

<p>This thesis reviews the experimental investigation of a non-smooth dynamical system consisting of two pendula; a large pendulum attached to a frame with an impact wall, and a small pendulum, which shares its axis of rotation with the large pendulum and can impact against the large pendulum. The system is forced with a sinusoidal horizontal motion, and due to the nonlinearities present in pendula as well as the discontinuous forcing from impacts, exhibits a wide range of behavior. Periodic, quasi-periodic, and chaotic responses all are possible, hysteresis is present, and grazing bifurcations allow for spontaneous change of behavior and the appearance of chaotic responses without following a traditional route to chaos. This thesis follows from existing non-linear dynamics research on forced pendula, impacting systems (such as a bouncing ball) and doubly impacting systems (ball bouncing on top of a bouncing ball).</p> / Thesis

We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.

In this work present the broad-band noise structure in the 2-60 keV data of Cataclysmic Variables (CVs) with Anomalous X-Ray Pulsars (AXPs) and Soft Gamma Repeaters (SGRs). We analyzed Rossi X-ray Timing Explorer (RXTE) PCA data and derived time series from 27 CVs, 4 AXPs and 1 SGR using the RXTE archive. In general, CVs of different types all show broad band noise which can be fitted with power laws, using exponentional cut-offs, and Lorentzians in a similar way to power spectral (noise) characteristics of X-ray Binaries (XRBs). In general terms the power spectra show a power law index of (-)1.2-2. A rather large scale flattening of the power spectra exits in nonmagnetic systems in the low to very low frequency range. We observe that in low and high states/outbursts the noise in the high frequency range and low frequency range is changed. CVs show considerably low frequency noise. In addition, we recovered several possible QPOs in the X-ray wavelengths from CVs mainly from Intermediate Polar systems. AXP and SGR sources which are thought to be powered by either magnetic decay or accretion show band limited noise in their low frequencies. We also correlated their equal time interval noise characteristic with their burst states and discovered that in the two AXPs (1E 2259+586, 1E 1048.1-5937) noise correlates with their bursts.

In this dissertation we present four papers as chapters. In Chapter 2, we extended the techniques used for the Klein-Gordon Chain by Iooss, Kirchgässner, James, and Sire, to chains with non-nearest neighbor interactions. We look for travelling waves by reducing the Klein-Gordon chain with second nearest neighbor interaction to an advance-delay equation. Then we reduce the equation to a finite dimensional center manifold for some parameter regimes. By using the normal form expansion on the center manifold we were able to prove the existence of three different types of travelling solutions for the Klein Gordon Chain: periodic, quasi-periodic and homoclinic to periodic orbits with exponentially small amplitude. In Chapter 3 we include numerical methods for computing quasi-periodic solutions. We developed very efficient algorithms to compute smooth quasiperiodic equilibrium states of models in 1-D statistical mechanics models allowing non-nearest neighbor interactions. If we discretize a hull function using N Fourier coefficients, the algorithms require O(N) storage and a Newton step for the equilibrium equation requires only O(N log(N)) arithmetic operations. This numerical methods give rise to a criterion for the breakdown of quasi-periodic solutions. This criterion is presented in Chapter 4. In Chapter 5, we justify rigorously the criterion in Chapter 4. The justification of the criterion uses both Numerical KAM algorithms and rigorous results. The hypotheses of the theorem concern bounds on the Sobolev norms of a hull function and can be verified rigorously by the computer. The argument works with small modifications in all cases where there is an a posteriori KAM theorem. / text

In this thesis, RXTE observations of 4U 1907+09 are presented. Timing analysis of these data sets have yielded quasi periodic oscillations (QPOs) at orbital phases corresponding to the two flares in every orbital period. Known continuous spin down trend and QPO behaviour at the flares strongly suggest that a transient accretion disk occurs at the flares. Our findings strongly suggested that neutron star passes through the equatorial wind of Be companion star. During these passages a transient disk forms around Be neutron star.

On utilise la théorie de Floquet-Lin pour des systèmes différentiels linéaires quasi- périodiques pour établir des résultats d'existence et d'unicité et de dépendance continue des systèmes différentiels non linéaires quasi-périodiques. Et dans un second temps on établit un résultat de réductibilité d'un système différentiel linéaire presque-périodique en un système différentiel linéaire triangulaire supérieur avec conservation du nombre des solutions presque-périodiques indépendantes. Ensuite, un résultat d’existence et d’unicité et de dépendance continue des systèmes différentiels non linéaires presque-périodiques par rapport au terme du contrôle. / We use a Floquet theory for quasi-periodic linear ordinary differential equations due to Zhensheng Lin to obtain results, of existence, unicity, continuous and differentiable dependence, on the quasi-periodic solutions of quasi-periodic nonlinear ordinary differential equations. in a second time we establish the reducibility of linear systems of almost periodic differential equations into upper triangular systems of a. p. differential equations. This is done while the number of independent a. p. solutions is conserved. We prove existence and uniqueness of a. p. solutions of a nonlinear system with an a. p. linear part. Also we prove the continuous dependence of a. p. solutions of a nonlinear system with respect to an a. p. control term.

La surveillance de la santé foetale permet aux cliniciens d’évaluer le bien-être du foetus,de faire une détection précoce des anomalies cardiaques foetales et de fournir les traitementsappropriés. Les développements technologies actuels visent à permettre la mesurede l’électrocardiogramme (ECG) foetal de façon non-invasive afin d’extraire non seulementle rythme cardiaque mais également la forme d’onde du signal. Cet objectif est rendudifficile par le faible rapport signal sur bruit des signaux mesurés sur l’abdomen maternel.Cette mesure est donc toujours un challenge auquel se confrontent beaucoup d’études quiproposent des solutions de traitement de signal basées sur la seule modalité ECG.Le but de cette thèse est d’utiliser la modélisation des processus Gaussiens pour améliorerl’extraction des signaux cardiaques foetaux, dans une base multi-modale. L’ECG est utiliséconjointement avec le signal Phonocardiogramme (PCG) qui peut apporter une informationcomplémentaire à l’ECG. Une méthode générale pour la modélisation des signauxquasi-périodiques est présentée avec l’application au débruitage de l’ECG et à l’extractionde l’ECG du foetus. Différents aspects de la multi-modalité (synchronisation, · · · ) proposéesont étudiées afin de détecter avec plus de robustesse les battements cardiaques foetaux.La méthode considère l’application sur les signaux ECG et PCG à travers deux aspects:l’aspect du traitement du signal et l’expérimental. La modélisation des processus Gaussien,avec le signal PCG pris comme la référence, est utilisée pour extraire des modèles flexibleset des estimations non linéaires de l’information. La méthode cherche également à faciliterla mise en oeuvre pratique en utilisant un codage 1-bit des signaux de référence.Le modèle proposé est validé sur des signaux synthétiques et également sur des donnéespréliminaires réelles qui ont été enregistrées afin d’amorcer la constitution d’une base dedonnées multi-modale synchronisée. Les premiers résultats montrent que la méthode permettraà terme aux cliniciens d’étudier les battements cardiaques ainsi que la morphologiede l’ECG. Ce dernier aspect était jusqu’à présent limité à l’analyse d’enregistrements ECGinvasifs prélevés pendant l’accouchement par le biais d’électrodes posées sur le scalp dufoetus. / Fetal health must be carefully monitored during pregnancy to detect early fetal cardiac diseases, and provide appropriate treatment. Technological development allows a monitoring during pregnancy using the non-invasive fetal electrocardiogram (ECG). Noninvasive fetal ECG is a method not only to detect fetal heart rate, but also to analyze the morphology of fetal ECG, which is now limited to analysis of the invasive ECG during delivery. However, the noninvasive fetal ECG recorded from the mother's abdomen is contaminated with several noise sources among which the maternal ECG is the most prominent.In the present study, the problem of noninvasive fetal ECG extraction is tackled using multi-modality. Beside ECG signal, this approach benefits from the Phonocardiogram (PCG) signal as another signal modality, which can provide complementary information about the fetal ECG.A general method for quasi-periodic signal analysis and modeling is first described and its application to ECG denoising and fetal ECG extraction is explained. Considering the difficulties caused by the synchronization of the two modalities, the event detection in the quasi-periodic signals is also studied which can be specified to the detection of the R-peaks in the ECG signal.The method considers both clinical and signal processing aspects of the application on ECG and PCG signals. These signals are introduced and their characteristics are explained. Then, using PCG signal as the reference, the Gaussian process modeling is employed to provide the possibility of flexible models as nonlinear estimations. The method also tries to facilitate the practical implementation of the device by using the less possible number of channels and also by using only 1-bit reference signal.The method is tested on synthetic data and also on real data that is recorded to provide a synchronous multi-modal data set.Since a standard agreement for the acquisition of these modalities is not yet taken into much consideration, the factors which influence the signals in recording procedure are introduced and their difficulties and effects are investigated.The results show that the multi-modal approach is efficient in the detection of R-peaks and so in the extraction of fetal heart rate, and it also provides the results about the morphology of fetal ECG.

Learning is a mechanism intrinsic to all sentient biological systems. Despite the diverse range of paradigms that exist, it appears that an artificial system has yet to be developed that can emulate learning with a comparable degree of accuracy or efficiency to the human brain. With the development of new approaches comes the opportunity to reduce this disparity in performance. A model presented by Janson and Marsden [arXiv:1107.0674 (2011)] (Memory foam model) redefines the critical features that an intelligent system should demonstrate. Rather than focussing on the topological constraints of the rigid neuron structure, the emphasis is placed on the on-line, unsupervised, classification, retention and recognition of stimuli. In contrast to traditional AI approaches, the system s memory is not plagued by spurious attractors or the curse of dimensionality. The ability to continuously learn, whilst simultaneously recognising aspects of a stimuli ensures that this model more closely embodies the operations occurring in the brain than many other AI approaches. Here we consider the pertinent deficiencies of classical artificial learning models before introducing and developing this memory foam self-shaping system. As this model is relatively new, its limitations are not yet apparent. These must be established by testing the model in various complex environments. Here we consider its ability to learn and recognize the RGB colours composing cartoons as observed via a web-camera. The self-shaping vector field of the system is shown to adjust its composition to reflect the distribution of three-dimensional inputs. The model builds a memory of its experiences and is shown to recognize unfamiliar colours by locating the most appropriate class with which to associate a stimuli. In addition, we discuss a method to map a three-dimensional RGB input onto a line spectrum of colours. The corresponding reduction of the models dimensions is shown to dramatically improve computational speed, however, the model is then restricted to a much smaller set of representable colours. This models prototype offers a gradient description of recognition, it is evident that a more complex, non-linear alternative may be used to better characterize the classes of the system. It is postulated that non-linear attractors may be utilized to convey the concept of hierarchy that relates the different classes of the system. We relate the dynamics of the van der Pol oscillator to this plastic self-shaping system, first demonstrating the recognition of stimuli with limit cycle trajectories. The location and frequency of each cycle is dependent on the topology of the systems energy potential. For a one-dimensional stimuli the dynamics are restricted to the cycle, the extension of the model to an N dimensional stimuli is approached via the coupling of N oscillators. Here we study systems of up to three mutually coupled oscillators and relate limit cycles, fixed points and quasi-periodic orbits to the recognition of stimuli.

Resting state (rs)fMRI allows measurement of brain functional connectivity and has identified default mode (DMN) and task positive (TPN) network disruptions as promising biomarkers for Alzheimer’s disease (AD). Quasi-periodic patterns (QPPs) of neural activity describe recurring spatiotemporal patterns that display DMN with TPN anti-correlation. We reasoned that QPPs could provide new insights into AD network dysfunction and improve disease diagnosis. We therefore used rsfMRI to investigate QPPs in old TG2576 mice, a model of amyloidosis, and age-matched controls. Multiple QPPs were determined and compared across groups. Using linear regression, we removed their contribution from the functional scans and assessed how they reflected functional connectivity. Lastly, we used elastic net regression to determine if QPPs improved disease classification. We present three prominent findings: (1) Compared to controls, TG2576 mice were marked by opposing neural dynamics in which DMN areas were anti-correlated and displayed diminished anti-correlation with the TPN. (2) QPPs reflected lowered DMN functional connectivity in TG2576 mice and revealed significantly decreased DMN-TPN anti-correlations. (3) QPP-derived measures significantly improved classification compared to conventional functional connectivity measures. Altogether, our findings provide insight into the neural dynamics of aberrant network connectivity in AD and indicate that QPPs might serve as a translational diagnostic tool.