Geometric Bifurcation Theory: Fisher Information Geometry Applied to Dynamic and Complex Systems

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.

Dr. Vinícius Barros is a physicist with a profound understanding and expertise in the area of statistical mechanics, dynamical systems and information geometry. He earned his Ph.D. in Applied Physics from São Paulo State University “Júlio de Mesquita Filho” (UNESP) in 2023, with a focus on topics intrinsically linked to information geometry and dynamical systems, including bifurcations. His solid academic background also includes a Master's degree (2018) and a Bachelor's degree in Physics (2016), both from UNESP. Dr. Barros's expertise encompasses several crucial areas for the geometric theory of bifurcations, such as dynamical systems, chaos theory, Fisher information geometry, differential geometry, and the Fisher and Rao metrics, as well as scalar curvature. His research has focused on applying tools from statistical physics and information geometry to gain deep insights into dynamical systems, with a particular interest in understanding and characterizing bifurcations through a geometric lens.

Dr. João Peres Vieira earned a bachelor’s degree in Mathematics from the São Carlos Federal University in 1984, a Master’s in Mathematics from the São Paulo University in 1988, and a Ph.D. in Mathematics from the same institution in 1995. In 2012, he achieved his Habilitation in Mathematics from São Paulo State University “Júlio de Mesquita Filho”, where he currently serves as an Associate Professor. With extensive expertise in Mathematics, Dr. Vieira specializes in algebraic topology and dynamical systems. His primary research interests focus on fixed points, coincidence theory, and their applications within topological dynamics, offering important insights into the behaviour and structure of complex dynamical systems. His contributions reflect a deep commitment to advancing the understanding of both theoretical and applied aspects of these mathematical fields.

Dr. Edson Denis Leonel is a Professor of Physics at São Paulo State University (UNESP), Rio Claro, Brazil. He has been working on scaling investigations since his Ph.D. in 2003, where he conducted the first study of scaling behavior in the chaotic sea of the Fermi-Ulam model. His research group has developed a variety of approaches and formalisms to investigate and characterize scaling properties across a wide range of systems, including one-dimensional mappings, ordinary differential equations, cellular automata, meme propagation, and time-dependent billiards. His group has investigated different types of transitions using scaling investigations, including but not limited to: (i) the transition from integrability to non-integrability; (ii) the transition from limited to unlimited diffusion; and (iii) the production and suppression of Fermi acceleration - the latter involving the analytical solution of the diffusion equation. Professor Leonel and his collaborators have published more than 190 scientific papers in respected international journals, including Physical Review Letters, Physics Reports, and many others. He is the author of Scaling Laws in Dynamical Systems (Springer & Higher Education Press, 2021), and Dynamical Phase Transitions in Chaotic Systems (Springer & Higher Education Press, 2023), Foundations of Statistical Mechanics (Generis, 2025) as well as two books in Portuguese: one on Statistical Mechanics (Blucher, 2015) and another on Nonlinear Dynamics (Blucher, 2019).