This thesis consists of six chapters and deals with four topics related to De Rham Theory on semialgebraic sets. The first topic deals with L-infinity cohomology on semialgebraic sets. We introduce smooth L-infinity differential forms on a singular (semialgebraic) space X in Rn. Roughly speaking, a smooth L-infinity differential form is a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and of bounded size (in the metric induced from Rn). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplex into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L-infinity forms. The second topic is a construction of a Lipschitz deformation retraction on a neighborhood of a point in a semialgebraic set with estimates on its derivatives. Such a deformation retraction is the key to the results of the first and the third topics. The third topic is related to Poincare inequality on a semialgebraic set. We study Poincare type Lp inequality for differential forms on a compact semialgebraic subset of Rn for p >> 1. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives from the second topic and then we extend it to a global inequality by employing a technique developed in the appendix. As a consequence we obtain an isomorphism between Lp cohomology and singular cohomology of a normal compact semialgebraic set. The final topic is in the appendix. It deals with an explicit proof of Poincare type inequality for differential forms on compact manifolds. We prove the latter inequality by means of a constructive 'globalization' method of a local Poincare inequality on convex sets. The appendix serves as a model case for the results of the third topic in Chapter 5.

This thesis consists of six chapters and deals with four topics related to De Rham Theory on semialgebraic sets. The first topic deals with L-infinity cohomology on semialgebraic sets. We introduce smooth L-infinity differential forms on a singular (semialgebraic) space X in Rn. Roughly speaking, a smooth L-infinity differential form is a collection of smooth forms on disjoint smooth subsets (stratification) of X with matching tangential components on the adjacent strata and of bounded size (in the metric induced from Rn). We identify the singular homology of X as the homology of the chain complex generated by semialgebraic singular simplices, i.e. continuous semialgebraic maps from the standard simplex into X. Singular cohomology of X is defined as the homology of the Hom dual to the chain complex of the singular chains. Finally, we prove a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth L-infinity forms. The second topic is a construction of a Lipschitz deformation retraction on a neighborhood of a point in a semialgebraic set with estimates on its derivatives. Such a deformation retraction is the key to the results of the first and the third topics. The third topic is related to Poincare inequality on a semialgebraic set. We study Poincare type Lp inequality for differential forms on a compact semialgebraic subset of Rn for p >> 1. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives from the second topic and then we extend it to a global inequality by employing a technique developed in the appendix. As a consequence we obtain an isomorphism between Lp cohomology and singular cohomology of a normal compact semialgebraic set. The final topic is in the appendix. It deals with an explicit proof of Poincare type inequality for differential forms on compact manifolds. We prove the latter inequality by means of a constructive 'globalization' method of a local Poincare inequality on convex sets. The appendix serves as a model case for the results of the third topic in Chapter 5.

When wind blows over a stratified lake an interface tilt is often generated, and internal waves usually appear after the wind stops. Internal waves in lakes are studied in many literatures, but most assume a hydrostatic pressure balance. In this thesis we discuss the internal Poincare waves and Kelvin waves in a rotating, continuously stratified, flat-bottom, circular lake with fully nonlinear and non-hydrostatic effects. An analytical solution is derived for the linearized system and it provides initial conditions used in the MIT General Circulation Model (MITgcm). This model is chosen due to its non-hydrostatic capability. Both Poincare waves and Kelvin waves are considered. The analytical solution of the linear system is verified numerically when the wave amplitude is small. As the wave amplitude increases the waves become more nonlinear. Poincare waves steepen and generate solitary-like waves with shorter wavelengths, but most of the energy contained in these waves is transferred back and forth between the parent wave and the solitary-like waves. Kelvin waves, on the other hand, steepen and lose their energy to solitary-like waves. The appearance of the solitary-like waves is not absolutely clear and higher resolution is required to clear up the details of this process. This conclusion agrees with de la Fuente et al (2008) who discussed the internal waves in a two-layer model. Moreover, in the Kelvin waves case, unexpected small waves are generated at the side boundaries and travel inwards. The wave amplitude and wavelength of these spurious waves become smaller as the horizontal resolution increases. One possible reason to explain these waves is the use of square grids to approximate the circular lake.

Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\'e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence.

This dissertation creates practical methods for Poincaré return mappings of individual and networked neuron models. Elliptic bursting models are found in numerous biological systems, including the external Globus Pallidus (GPe) section of the brain; the focus for studies of epileptic seizures and Parkinson's disease. However, the bifurcation structure for changes in dynamics remains incomplete. This dissertation develops computer-assisted Poincaré ́maps for mathematical and biologically relevant elliptic bursting neuron models and central pattern generators (CPGs). The first method, used for individual neurons, offers the advantage of an entire family of computationally smooth and complete mappings, which can explain all of the systems dynamical transitions. A complete bifurcation analysis was performed detailing the mechanisms for the transitions from tonic spiking to quiescence in elliptic bursters. A previously unknown, unstable torus bifurcation was found to give rise to small amplitude oscillations. The focus of the dissertation shifts from individual neuron models to small networks of neuron models, particularly 3-cell CPGs. A CPG is a small network which is able to produce specific phasic relationships between the cells. The output rhythms represent a number of biologically observable actions, i.e. walking or running gates. A 2-dimensional map is derived from the CPGs phase-lags. The cells are endogenously bursting neuron models mutually coupled with reciprocal inhibitory connections using the fast threshold synaptic paradigm. The mappings generate clear explanations for rhythmic outcomes, as well as basins of attraction for specific rhythms and possible mechanisms for switching between rhythms.

When wind blows over a stratified lake an interface tilt is often generated, and internal waves usually appear after the wind stops. Internal waves in lakes are studied in many literatures, but most assume a hydrostatic pressure balance. In this thesis we discuss the internal Poincare waves and Kelvin waves in a rotating, continuously stratified, flat-bottom, circular lake with fully nonlinear and non-hydrostatic effects. An analytical solution is derived for the linearized system and it provides initial conditions used in the MIT General Circulation Model (MITgcm). This model is chosen due to its non-hydrostatic capability. Both Poincare waves and Kelvin waves are considered. The analytical solution of the linear system is verified numerically when the wave amplitude is small. As the wave amplitude increases the waves become more nonlinear. Poincare waves steepen and generate solitary-like waves with shorter wavelengths, but most of the energy contained in these waves is transferred back and forth between the parent wave and the solitary-like waves. Kelvin waves, on the other hand, steepen and lose their energy to solitary-like waves. The appearance of the solitary-like waves is not absolutely clear and higher resolution is required to clear up the details of this process. This conclusion agrees with de la Fuente et al (2008) who discussed the internal waves in a two-layer model. Moreover, in the Kelvin waves case, unexpected small waves are generated at the side boundaries and travel inwards. The wave amplitude and wavelength of these spurious waves become smaller as the horizontal resolution increases. One possible reason to explain these waves is the use of square grids to approximate the circular lake.

The spatial structure and temporal evolution of sea breeze and the latitudinal distribution of propagation and mixing of sea breeze driven near-inertial ocean response in the Gulf of Mexico are investigated using comprehensive data sets and a non-linear numerical model. Near 30�N, inertial oceanic response is significantly enhanced by a near-resonant condition between inertial and diurnal forcing frequencies. Observational results indicate that sea breeze variability peaks in summer and extends at least 300 km offshore with continuous seaward phase propagation. The maximum near-inertial oceanic response occurs in June when there is a shallow mixed layer, strong stratification, and an approximately 10-day period of continuous sea breeze forcing. Near-inertial current variance decreases in July and August due to the deepening of the mixed layer and a more variable phase relationship between the wind and current. River discharge varies interannually and can significantly alter the oceanic response during summer. During 1993, the ?great flood? of the Mississippi River deepens the summer mixed layer and reduces the sea breeze response. The near-inertial currents can provide considerable vertical mixing on the shelf in summer, as seen by the suppression of bulk Richardson number during strong near-inertial events. Three-dimensional idealized simulations show that the coastal oceanic response to sea breeze is trapped poleward of 30� latitude, however, it can propagate offshore as Poincare waves equatorward of 30� latitude. Near 30� latitude, the maximum oceanic response to sea breeze moves offshore slowly because of the near-zero group speed of Poincare waves at this latitude. The lateral energy flux convergence plus the energy input from the wind is maximum near the critical latitude, leading to increased vertical mixing. This local dissipation is greatly reduced at other latitudes. Simulations with realistic bathymetry of the Gulf of Mexico confirm that a basin-wide ocean response to coastal sea breeze forcing is established in the form of Poincare waves. This enhanced vertical mixing is consistent with observations on the Texas-Louisiana Shelf. Comparison of the three-dimensional and one-dimensional models shows some significant limitations of one-dimensional simplified models for sea breeze simulations near the critical latitude.

Periodic and chaotic dynamical systems follow deterministic equations such as Newton's laws of motion. To distinguish the difference between two systems, the initial conditions have an important role. Chaotic behaviors or dynamics are characterized by sensitivity to initial conditions. Mathematically, a chaotic system is defined as a system very sensitive to initial conditions. A small difference in initial conditions causes unpredictability in the final outcome. If error is measured from the initial state, the relative error grows exponentially. Prediction becomes impossible and finally, chaotic systems can come to become stochastic system. To make chaotic motion, the number of variables in the system should be above three and there should be non-linear terms coupling several of the variables in the equation of motion. Phase space is defined as the space spanned by the coordinate and velocity vectors. In our case, mixing zone is phase space. With the above characteristics - the initial condition sensitivity of a chaotic system, our plan is to find most efficient chaotic stirrer. In this thesis, we present four methods to measure mixing state based on the chaotic dynamics theory. The Lyapunov exponent is a measure of the sensitivity to initial conditions and can be used to calculate chaotic strength. We can decide the chaotic state with one real number and measure efficiency of the chaotic mixer and find the optimum frequency. The Poincare section method provides a means for viewing the phase space diagram so that the motion is observed periodically. To do this, the trajectory is sectioned at regular intervals. With the Poincare section method, we can find 'islands' considered as bad mixed zones so that the mixing state can be measured qualitatively. With the chaotic dynamics theory, the initial length of the interface can grow exponentially in a chaotic system. We will show the above characteristics of the chaotic system to prove as fact that our model is an efficient chaotic mixer. The final goal for making chaotic stirrer is how to implement efficient dispersed particles. The box counting method is focused on measurement of the particles dispersing state. We use snap shots of the mixing process and with these snap shots, we devise a plan to measure particles' dispersing rate using the box-counting method.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

Made available in DSpace on 2015-03-03T11:52:41Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-10Bitstream added on 2015-03-03T12:07:11Z : No. of bitstreams: 1 000803634.pdf: 508680 bytes, checksum: bab04e8b7f98eda0b0883ef25409ac2f (MD5) / Este trabalho está dividido em duas partes. Na primeira, iremos introduzir a nomenclatura de Fillipov e os conceitos básicos e em seguida iremos estudar a classificação de centros em sistemas lineares por partes em duas zonas no plano. Para tal fim, iremos encontrar uma mudança de variáveis que nos permita reduzir o número de parâmetros de doze para cinco. Na segunda parte deste trabalho iremos estudar o surgimento de ciclos limites para esta classe de campo de vetores descontínuos através das aplicações de Poincaré em cada zona. Neste trabalho nos restringiremos ao caso em que não há regiões de sliding no conjunto de descontinuidade / This work is divided into two parts. At rst part we introduce the nomenclature of Fillipov and the basics concepts and then we will study the classi cation of centers in piecewise linear systems in the plan. To this end we nd a change of variables that allows us to reduce the initial twelve parameters to ve. In the second part of this work we study the emergence of limit cycles for this class of systems through the Poincar e applications in each region of the plan. In this work we will consider only the case where the set of discontinuity has no sliding regions