Discrete Dynamical Systems in Low Dimensions

BSc Thesis (Mathematics) — University of Extremadura, 2025

This thesis was proposed and supervised by José Luis Bravo Trinidad, full professor of mathematical analysis at the University of Extremadura. It studies how discrete dynamical systems arise naturally from differential equations: discretizing an ODE via Euler's method (with Newton's method as a particular case) produces a difference equation whose long-term behaviour reflects properties of the original equation. The first part analyzes the stability of fixed points through linearization and Taylor expansions, and includes a classification of the stability of non-hyperbolic fixed points (f'(x̂) = ±1) in terms of higher-order derivatives, derived and proved independently, introducing one-sided notions of asymptotic stability. The second part extends the analysis to perturbations of monotone systems, covering hyperbolic and degenerate fixed points and period-doubling bifurcations. The final chapter is a walkthrough of chaos: symbolic dynamics, the itinerary function and its conjugacy with the shift map, Sharkovskii's theorem, and the classical proof that the logistic family is chaotic for λ > 4. Most results follow Hale and Koçak's Dynamics and Bifurcations (Springer, 1991), with proofs elaborated and completed where the original arguments are terse or incomplete.