Edson Denis Leonel - Böcker

This book discusses many of the common scaling properties observed in some nonlinear dynamical systems mostly described by mappings.

This book discusses many of the common scaling properties observed in some nonlinear dynamical systems mostly described by mappings.

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.

This book discusses some scaling properties and characterizes two-phase transitions for chaotic dynamics in nonlinear systems described by mappings. The chaotic dynamics is determined by the unpredictability of the time evolution of two very close initial conditions in the phase space. It yields in an exponential divergence from each other as time passes. The chaotic diffusion is investigated, leading to a scaling invariance, a characteristic of a continuous phase transition. Two different types of transitions are considered in the book. One of them considers a transition from integrability to non-integrability observed in a two-dimensional, nonlinear, and area-preserving mapping, hence a conservative dynamics, in the variables action and angle. The other transition considers too the dynamics given by the use of nonlinear mappings and describes a suppression of the unlimited chaotic diffusion for a dissipative standard mapping and an equivalent transition in the suppression of Fermi acceleration in time-dependent billiards. This book allows the readers to understand some of the applicability of scaling theory to phase transitions and other critical dynamics commonly observed in nonlinear systems. That includes a transition from integrability to non-integrability and a transition from limited to unlimited diffusion, and that may also be applied to diffusion in energy, hence in Fermi acceleration. The latter is a hot topic investigated in billiard dynamics that led to many important publications in the last few years. It is a good reference book for senior- or graduate-level students or researchers in dynamical systems and control engineering, mathematics, physics, mechanical and electrical engineering.