Chaotic Dynamics in the Cheeger Problem: A Focus on Rectangular Geometries

Abstract

This paper presents a novel investigation into the dynamics of the Cheeger problem for rectangular domains. The study introduces a new framework by defining the dynamical system associated with the classical Cheeger problem specifically for rectangular shapes. A key finding is that the conditions m < 0 and q > m are sufficient for chaotic behavior in the dynamical system defined for the Cheeger problem on rectangles. In this context, b(t) = e^mt represents one of the time-varying dimensions of the rectangle, and εb(t) = e^qt represents the perturbation in the same dimension. The maximal deterministic Lyapunov exponent is rigorously derived, showing that chaos emerges as one rectangle dimension decays while perturbations grow. Numerical illustrations confirm chaos onset at finite times t = 0.2383 and t = 0.2494, demonstrating exponential divergence of trajectories and sensitive dependence on initial conditions. These results establishing fundamental limits on the predictability horizon and enabling precise timing for interventions, thereby bridging theoretical insights with practical foresight and control. Sensitivity analysis further reveals that increasing q accelerates chaos onset, while increasing m stabilizes the system. Altogether, the work advances both theoretical understanding and practical insights into geometric dynamics for complex systems.