Global minimization of Hopf bifurcation surfaces with application to nematic electroconvection

Abstract

This dissertation addresses two problems which frequently arise in applications: locating the Hopf bifurcations for autonomous systems of ordinary differential equations and finding the global minimum of continuous functions. These problems are important in dynamical systems and optimization, respectively. The first problem is to locate Hopf bifurcation points of dynamical systems. Two approaches are used to find the Hopf points. The first approach is the polynomial resultants method. In this method, the characteristic polynomial of a special matrix is introduced and then a necessary and sufficient condition for this polynomial to have two purely imaginary roots is given. The second approach is the Werner Method. In this method, bordered matrices are used to classify the set of matrices with rank deficiency two, at which Hopf points occur. The second problem is to compute the global minimum of a continuous function defined on a compact region. We use two approaches to find the global minimum. The first approach is the Nelder-Mead method. This method attempts to find the minimum of a continuous function using only functions values without using any information about derivatives. The second approach is the Cell Exclusion Algorithm. We begin with some region on which we wish to determine the global minimum. If a cell fails the minimization condition, it is discarded. If a cell satisfies the condition, it is subdivided and the condition is then applied to the new cells. For the minimization condition, we choose the monotonicity condition and we then introduce the space of functions of bounded variation which contains all functions that have finite arclength in 1-dimension. We consider the Jordan Decomposition Theorem and use it in the monotonicity condition. Finally, we apply those methods to the mathematical model of the electroconvection in nematic liquid crystals. The surface to be minimized consists of Hopf bifurcation points and comes from a linear stability analysis performed on the weak electrolyte model and is used to test previous algorithms and compare them. In this dissertation we present the theory behind Hopf bifurcation and global optimization. We present both types of algorithms and give multiple numerical examples.