Chaos Detection in the Logistic Map Using Lyapunov Exponents

Abstract

This paper investigates the transition to chaos in the logistic map, a fundamental model in nonlinear dynamics, by employing Lyapunov exponents as a quantitative measure. We begin by establishing the mathematical framework of one-dimensional maps and the formal definition of the Lyapunov exponent as an indicator of sensitive dependence on initial conditions. A numerical method for computing the exponent is implemented and applied to the logistic map across a range of its control parameter. The resulting Lyapunov spectrum is then systematically compared to the map's bifurcation diagram, demonstrating how positive Lyapunov exponents precisely correspond to chaotic regimes. The analysis highlights the exponent's utility not only as a binary chaos indicator but also as a measure of chaos intensity. This study reaffirms the Lyapunov exponent as a robust and essential tool for characterizing dynamical systems.