Exploring Non-linear Dynamics: Constructing the Bifurcation Diagram of a Damped Driven Pendulum using Python

In this paper, we present a detailed analysis and construction of the bifurcation diagram for the damped-driven pendulum system. The bifurcation diagram in general presents the qualitative changes of the steady-state behavior for the pendulum. For this purpose, we implement the use of the Python programming language with the inclusion of scientific libraries. This nonlinear dynamical system is an example of a system that exhibits a chaotic regime, which is the sensitivity of its behavior to the initial conditions and the parameters of the system. We investigate the response of the system to a range of drive strengths γ applied. By changing the driving strength, the system reveals patterns of periodicity, quasi-periodic, and chaotic regimes. The critical values where it goes from regular motion to chaotic one are highlighted, offering further understanding of the mechanisms of the transitions. This study presents the use of the Python programming language for the modeling and visualization of non-dynamic systems and contributes to a deeper understanding of nonlinear oscillator dynamics.