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.

Autorentext

Ibraheem Alolyan is an associate professor of mathematics at King Saud University, Saudi Arabia. He got his PhD in mathematics from Colorado State University, USA. His research interests include numerical analysis and optimization. He has published papers in optimization and wrote a book in Linear Programming.E-mail: ialolyan@ksu.edu.sa

Klappentext

This book addresses two problems which frequently arise in applications:locating the Hopf bifurcations for autonomous systems of ordinary differentialequations and finding the global minimum of continuous functions. Theseproblems 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 theWerner Method. The second problem is to compute the global minimum of a continuous functiondefined on a compact region. We use two approaches to find the global minimum; the Nelder-Mead method and the Cell ExclusionAlgorithm. Finally, we apply those methods to the mathematical model of the electroconvectionin nematic liquid crystals. The surface to be minimized consists of Hopfbifurcation points and comes from a linear stability analysis performed on the weakelectrolyte model and is used to test previous algorithms and compare them.In this book, we present the theory behind Hopf bifurcation and globaloptimization and give multipleexamples.