O sistema de Lorenz e a origem do caos em sistemas determinísticos

Abstract

This monograph presents an in-depth study of the Lorenz system, integrating theoretical foundations, qualitative analysis, stability analysis, and a numerical investigation of chaotic behavior. Initially, the essential concepts of dynamical systems—equilibrium points, linearization, bifurcations, and Lyapunov’s Second Method—are reviewed, providing the basis for understanding higher-dimensional nonlinear systems. Next, a detailed analysis of the Lorenz System is developed, including its physical properties related to thermal convection, the Rayleigh and Prandtl parameters, the system’s symmetries, and volume contraction. The stability of the equilibrium points is examined, with particular emphasis on the Hopf bifurcation that precedes the chaotic regime. Through numerical simulations using a high-order Runge–Kutta method, the strange attractor is visualized, highlighting sensitivity to initial conditions and the fractal structure of trajectories. The work thus consolidates the understanding of the transition from regular behavior to deterministic chaos, characterizing the Lorenz System as a landmark in modern dynamical systems theory.