Geometric bifurcation theory: Fisher information geometry applied to dynamical and complex systems

Although geometrical methods have been applied across many fields of physics, their systematic use in connection with bifurcations and nonlinear phenomena only emerged between 2019 and 2024, when Fisher information geometry was employed to study bifurcations, limit cycles, and other nonlinear dynamical phenomena, leading to the covariant formulation of geometric bifurcation theory (GBT). In this approach, incorporating Fisher information theory into the axioms of nonlinear dynamics yields a corresponding Riemannian metric, allowing for the representation of dynamical systems as Riemannian manifolds. The metric and its scalar curvature are particularly valuable tools for exploring nonlinear phenomena, especially in cases where standard methods provide limited or no solutions. This report aims to review the main contributions of this geometrical formalism within the framework of dynamical systems governed by differential equations. In particular, we present a detailed overview of the mathematical framework of GBT, including the construction of Riemannian manifolds from dynamical systems, the Fisher information metric, and the role of scalar curvature in detecting local and global bifurcations, limit cycles, and other nonlinear phenomena. We also discuss how GBT provides a solution to the second part of Hilbert’s sixteenth problem, and how this result aligns with earlier findings obtained some time ago through other methods under somewhat different circumstances. Finally, we highlight the current state of GBT and promising directions for future research.