Geometric instability of the nonlinear financial system

To understand the complexity of financial market, this study investigates the geometric instability of nonlinear financial systems. We utilize the Kosambi–Cartan–Chern (KCC) theory to transform the three-dimensional financial system into a second-order differential equation (SODE), and characterize its geometric structure with KCC invariants. It is demonstrated that the boundary equilibrium is Jacobi unstable, and the symmetric pair of equilibria are Jacobi conditionally stable. Our innovative probabilistic analysis reveals that Jacobi stability manifests with an extremely low probability across an economically plausible parameter space. The Jacobi instability shows that the system’s trajectories are geometrically sensitive. This demonstrates that the financial system can be geometrically fragile despite being Lyapunov stable, introducing a novel layer of instability rooted in the deviation curvature tensor. Hence, the KCC theory is an essential tool for future stability analysis in financial markets; this work bridges a critical gap between geometric stability theory and financial markets.