Geometric Bifurcation Theory

This book will present a comprehensive exploration of the geometric bifurcation theory (GBT), a novel approach that employs information geometry to analyze dynamical systems. It will delve into the mathematical foundations of GBT, including the Riemannian metrical structure of parameter spaces, Fisher information metric, scalar curvature, and their application to local and global bifurcations. The book will cover the limitations of classical bifurcation theory (CBT) and demonstrate how GBT overcomes these by providing a more complete characterization of stability and addressing the global behavior of nonlinear dynamical systems. Specific topics will include the geometric interpretation of bifurcations, stability analysis using curvature scalar, scaling analysis using Fisher information, and the application of GBT to study complex and nonlinear phenomena, especially where the standard methods show little or no solution.