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The nonlinear dynamic analysis of aeroelastic systems is a topic that has been covered extensively in the literature. The two main sources of nonlinearities in such systems, structural and aerodynamic nonlinearities, have analyzed numerically, analytically and experimentally. In this research project, the aerodynamic nonlinearity arising from the stall behavior of an airfoil is analyzed. Experimental data was used to fit a piecewise linear curve to describe the lift versus angle of attack behavior for a NACA 0012 2 DOF airfoil. The piecewise linear system equilibrium points are found and their stability analyzed. Bifurcations of the equilibrium points are analyzed and applying continuation software the bifurcation diagrams of the system are shown. Border collision and rapid/Hopf bifurcations are the two main bifurcations of the system equilibrium points. Chaotic behavior represented in the intermittent route to chaos was also observed and shown as part of the system dynamic analysis. Finally, sets of initial conditions associated with the system behavior are defined. Numerical simulations are used to show those sets, their subsets and their behavior with respect to the system dynamics. Poincaré sections are produced for both the periodic and the chaotic solutions of the system. The proposed piecewise linear model introduced some interesting dynamics for such systems. The introduction of the border collision bifurcation and the existence of periodic and chaotic solutions for the system are some examples. The model also enables the understanding of the mapping of initial conditions as it defines clear boundaries with different dynamics that can be used as Poincaré sections to understand further the global system dynamics. One of the constraints of the system is its validity as it is dependent on the range of the experimental data used to generate the model. This can be addressed by adding more linear pieces to the system to cover a wider range of the dynamics.

The starting point of the research developed in this thesis was the work done by Arek Goetz in his PhD thesis [Dynamics of Piecewise Isometrics, University of Illinois, 1996). Following his dissertation, we have considered the simple example of a piecewise rotation in two convex atoms defined in the whole plane (now commonly known as the Goetz map) as our main source of motivation. The first main achievement of our work was the construction of a new family of polygonal sets which are, in fact, global attractors. These examples are very similar in nature to the Sierpinski-gasket triangle presented in Goetz [1998c]. The next natural step was to argue that the definition of a piecewise isometric attractor is not entirely suitable in this context due to the lack of uniqueness. Under a new definition of attractor, some results are proved involving the properties of quasi-invariance and regularity existing in all examples available in the literature on the Goetz map. Following that, we attempt to generalise the results of Goetz regarding periodic cells and periodic points to unbounded spaces. We prove that there is a fundamental discrepancy between piecewise rotations in odd and even dimensions. In the odd-dimensional case the existence of periodic points is rare; hence those that exist must be unstable under almost all perturbations, whereas in even dimensions periodic points are stable for a prevalent set of piecewise rotations. Furthermore, if a piecewise rotation is such that the free monoid generated by the linear parts of the induced rotations does not contain the identity map then it follows trivially that all diverging orbits must be irrationally coded. This implies, together with a result on the coding of the open connected components in the complement of the closure of the exceptional set, that there exist examples of piecewise isometries possessing irrational cells with positive measure. This scenario was not considered previously in the results proved by Goetz. Using a common recurrence argument and Goetz's characterisation of the closure of the exceptional set (see for instance, Goetz [2001]) we prove that every recurrent point (i.e., such that w(x) ? ?) must be rationally coded. Given an invertible piecewise isometry in a compact space we also show that the closure of the exceptional set equals that of its inverse. This sustains the common idea that forward and backward iterates of the discontinuity yield similar graphics. In the context of the Goetz map we have also investigated the appearance of symmetric patterns when plotting the closure of the exceptional set. Although the Goetz map is by its nature discontinuous, the existence of symmetry is still possible under the broader framework of almost-everywhere-symmetry. Finally, we briefly note that the local symmetry properties of symmetric patterns arising in the invertible Goetz map are in part due to the existence of a reversing-symmetry, which generates piecewise continuous reversing symmetries under iteration of the original Goetz map.

In this thesis, we discuss the development, implementation and testing of a piecewise linear (PWL) continuous Galerkin finite element method applied to the threedimensional diffusion equation. This discretization is particularly interesting because it discretizes the diffusion equation on an arbitrary polyhedral mesh. We implemented our method in the KULL software package being developed at Lawrence Livermore National Laboratory. This code previously utilized Palmer's method as its diffusion solver, which is a finite volume method that can produce an asymmetric coefficient matrix. We show that the PWL method produces a symmetric positive definite coefficient matrix that can be solved more efficiently, while retaining the accuracy and robustness of Palmer's method. Furthermore, we show that in most cases Palmer's method is actually a non-Galerkin PWL finite element method. Because the PWL method is a Galerkin finite element method, it has a firm theoretical background to draw from. We have shown that the PWL method is a well-posed discrete problem with a second-order convergence rate. We have also performed a simple mode analysis on the PWL method and Palmer's method to compare the accuracy of each method for a certain class of problems. Finally, we have run a series of numerical tests to uncover more properties of both the PWL method and Palmer's method. These numerical results indicate that the PWL method, partially due to its symmetric matrix, is able to solve large-scale diffusion problems very efficiently.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 106-108). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Oregon State University, 1996. / Typescript (photocopy). Includes bibliographical references (leaves 68-69). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1973. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

This dissertation applies policy improvement and successive approximation or value iteration to a general class of Markov decision processes with discounted costs. In particular, a class of Markov decision processes, called piecewise-linear, is studied. Piecewise-linear processes are characterized by the property that the value function of a process observed for one period and then terminated is piecewise-linear if the terminal reward function is piecewise-linear. Partially observable Markov decision processes have this property. It is shown that there are e-optimal piecewise-linear value functions and piecewise-constant policies which are simple. Simple means that there are only finitely many pieces, each of which is defined on a convex polyhedral set. Algorithms based on policy improvement and successive approximation are developed to compute simple approximations to an optimal policy and the optimal value function. / Business, Sauder School of / Graduate

The piecewise linear group is the set of all piecewise linear orientation preserving homeomorphisms from the interval to itself under the operation of composition. We present here a complete set of invariants to classify the conjugacy classes of this group. Our approach to this problem relies on the factorization of elements into elements having only a single breakpoint.

Piecewise-smooth stochastic systems have attracted a lot of interest in the last decades in engineering science and mathematics. Many investigations have focused only on one-dimensional problems. This thesis deals with simple two-dimensional piecewise-smooth stochastic systems in the absence of detailed balance. We investigate the simplest example of such a system, which is a pure dry friction model subjected to coloured Gaussian noise. The nite correlation time of the noise establishes an additional dimension in the phase space and gives rise to a non-vanishing probability current. Our investigation focuses on stick-slip transitions, which can be related to a critical value of the noise correlation time. Analytical insight is provided by applying the uni ed coloured noise approximation. Afterwards, we extend our previous model by adding viscous friction and a constant force. Then we perform a similar analysis as for the pure dry friction case. With parameter values close to the deterministic stick-slip transition, we observe a non-monotonic behaviour of the probability of sticking by increasing the correlation time of the noise. As the eigenvalue spectrum is not accessible for the systems with coloured noise, we consider the eigenvalue problem of a dry friction model with displacement, velocity and Gaussian white noise. By imposing periodic boundary conditions on the displacement and using a Fourier ansatz, we can derive an eigenvalue equation, which has a similar form in comparison to the known one-dimensional problem for the velocity only. The eigenvalue analysis is done for the case without a constant force and with a constant force separately. Finally, we conclude our ndings and provide an outlook on related open problems.