GLOBAL MINIMIZATION OF HOPF BIFURCATION SURFACES

This book 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 §polynomial resultants method and the§Werner Method. §§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 Nelder-Mead method and the Cell Exclusion§Algorithm. §§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 book, we present the theory behind Hopf §bifurcation and global§optimization and give multiple§examples.