Complex oscillations with multiple timescales - Application to neuronal dynamics

The results gathered in this thesis deal with multiple time scale dynamical systems near non-hyperbolic points, giving rise to canard-type solutions, in systems of dimension 2, 3 and 4. Bifurcation theory and numerical continuation methods adapted for such systems are used to analyse canard cycles as well as canard-induced complex oscillations in three-dimensional systems. Two families of such complex oscillations are considered: mixed-mode oscillations (MMOs) in systems with two slow variables, and bursting oscillations in systems with two fast variables. In the last chapter, we present recent results on systems with two slow and two fast variables, where both MMO-type dynamics and bursting-type dynamics can arise and where complex oscillations are also organised by canard solutions. The main application area that we consider here is that of neuroscience, more precisely low-dimensional point models of neurons displaying both sub-threshold and spiking behaviour. We focus on analysing how canard objects allow to control the oscillatory patterns observed in these neuron models, in particular the crossings of excitability thresholds

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations

Bifurcation results for periodic third-order Ambrosetti-Prodi-type problems

This paper presents sufficient conditions for the existence of a bifurcation point for nonlinear periodic third-order fully differential equations. In short, the main discussion on the parameter s about the existence, non-existence, or the multiplicity of solutions, states that there are some critical numbers σ0 and σ1 such that the problem has no solution, at least one or at least two solutions if ss>σ1, respectively, or with reversed inequalities. The main tool is the different speed of variation between the variables, together with a new type of (strict) lower and upper solutions, not necessarily ordered. The arguments are based in the Leray–Schauder’s topological degree theory. An example suggests a technique to estimate for the critical values σ0 and σ1 of the parameter

Time-Delay Switch Attack on Networked Control Systems, Effects and Countermeasures

In recent years, the security of networked control systems (NCSs) has been an important challenge for many researchers. Although the security schemes for networked control systems have advanced in the past several years, there have been many acknowledged cyber attacks. As a result, this dissertation proposes the use of a novel time-delay switch (TDS) attack by introducing time delays into the dynamics of NCSs. Such an attack has devastating effects on NCSs if prevention techniques and countermeasures are not considered in the design of these systems. To overcome the stability issue caused by TDS attacks, this dissertation proposes a new detector to track TDS attacks in real time. This method relies on an estimator that will estimate and track time delays introduced by a hacker. Once a detector obtains the maximum tolerable time delay of a plant’s optimal controller (for which the plant remains secure and stable), it issues an alarm signal and directs the system to its alarm state. In the alarm state, the plant operates under the control of an emergency controller that can be local or networked to the plant and remains in this stable mode until the networked control system state is restored. In another effort, this dissertation evaluates different control methods to find out which one is more stable when under a TDS attack than others. Also, a novel, simple and effective controller is proposed to thwart TDS attacks on the sensing loop (SL). The modified controller controls the system under a TDS attack. Also, the time-delay estimator will track time delays introduced by a hacker using a modified model reference-based control with an indirect supervisor and a modified least mean square (LMS) minimization technique. Furthermore, here, the demonstration proves that the cryptographic solutions are ineffective in the recovery from TDS attacks. A cryptography-free TDS recovery (CF-TDSR) communication protocol enhancement is introduced to leverage the adaptive channel redundancy techniques, along with a novel state estimator to detect and assist in the recovery of the destabilizing effects of TDS attacks. The conclusion shows how the CF-TDSR ensures the control stability of linear time invariant systems

Efficient Optimization Algorithms for Nonlinear Data Analysis

Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems. Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data. The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated. Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.Siirretty Doriast

Regularização e conjuntos minimais para sistemas dinâmicos não suaves

Orientadores: Marco Antonio Teixeira, Jaume Llibre SaloTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Os problemas discutidos nesta tese concentram-se principalmente na teoria dos sistemas dinâmicos não diferenciáveis, da qual vários tópicos são abordados. Os resultados principais podem ser resumidos da seguinte forma. Primeiramente, relaxa-se as hipóteses dos teoremas clássicos da teoria "averaging" para o cálculo de soluções periódicas de sistemas dinâmicos não diferenciáveis. Em segundo lugar, com relação a sistemas dinâmicos planares lineares por partes com duas zonas, mostra-se que ao oscilar a linha de descontinuidade obtém-se diferentes configurações de ciclos limite. Em particular, prova-se que para um dado número natural n existe um sistema dinâmico planar linear por partes com duas zonas tendo exatamente n ciclos limite. Além disso, usando a teoria de Chebyshev, fica estabelecido limites superiores ótimos para o número máximo de ciclos limites que algumas classes de sistemas dinâmicos planares lineares por partes com duas zonas podem ter quando o conjunto de descontinuidade é uma linha reta. Em terceiro lugar, introduz-se, no contexto de sistemas de Filippov, o conceito de órbita de Shilnikov deslizante e, em seguida, considera-se o problema Shilnikov para este caso. Por fim, estuda-se as recentes extensões das convenções de Filippov para soluções de sistemas dinâmicos descontínuos, obtendo-se resultados referentes a regularização e "pinching" no contexto destas novas convençõesAbstract: The problems discussed in this thesis focuses mainly in the theory of nonsmooth differential system. Several topics of this subject are treated. The main results may be resumed as following. First, the hypotheses of the classical averaging theorems are relaxed to compute periodic solutions of nonsmooth differential systems. Second, regarding planar piecewise linear differential system with two zones it is shown that oscillating the line of discontinuity several configurations of limit cycles can be obtained. In addition it is proved that for a given natural number n there exists a planar piecewise linear differential system with two zones having exactly n limit cycles. Moreover, using the Chebyshev theory, it is established sharp upper bounds for the maximum number of limit cycles that some classes of planar piecewise linear differential systems with two zones can have when the set of discontinuity is a straight line. Third, the concept of sliding Shilnikov orbit is introduced in the context of Filippov systems, then the Shilnikov problem is considered for this case. Finally, the recent extensions of the Filippov's conventions for solutions of discontinuous differential systems is studied and some results concerning its regularization are established. Moreover the pinching of continuous systems is studied in the context of these new conventionsDoutoradoMatematicaDoutor em Matemática2012/10231-7CAPESFAPES