A computational and geometric analysis of economic dynamics using Pontryagin’s maximum principle: advanced numerical simulations perspective

Abstract

This research develops and implements a comprehensive Python-based computational framework for solving optimal control problems in economic dynamics using Pontryagin’s Maximum Principle (PMP). We employ advanced numerical methods including shooting algorithms with adaptive ODE solving (RK45 method with automatic step-size control), robust optimization techniques with multiple starting points, and parallel computation for sensitivity analysis. The framework generates interactive 2D and 3D visualizations using Matplotlib and Plotly to reveal geometric properties of optimal trajectories. The simulations demonstrate exceptional numerical accuracy with convergence errors near zero ($$<10^{-15}$$) for well-behaved scenarios while identifying computational challenges in extreme parameter regimes. Key quantitative findings include: (1) precise convergence times to steady state (52.5 periods for baseline), (2) geometric curvature metrics ($$\kappa _{\max } = 0.187$$, $$\bar{\kappa } = 0.089$$, curvature index $$CI = 2.10$$, $$\tau _{\text {total}} = 2.34$$ radians), and (3) sensitivity measures showing significant numerical challenges in high parameter regimes (errors up to 3614.14 for $$\rho =0.04$$). Our comparative analysis shows advantages over traditional shooting methods through adaptive step control and integrated visualization. This work establishes computational implementation and advanced geometric visualization as essential complements to theoretical analysis in economic dynamics, offering broad implications for sustainable economic research and policy applications.