From linear to non-linear/chaotic pendulum: a computational study

Abstract

In this work, we have used computational techniques to examine how the dynamics of a simple pendulum change from linear to non-linear and chaotic. The graph of phase space, angular displacement versus time, and angular velocity versus time are thoroughly examined in our analysis. A significant shift is seen in these representations, particularly in the graph of angular displacement versus time and angular velocity versus time. As the non-linearity is enhanced, we see a progressive movement from circular to oval shapes in phase space. In damped and forced pendulum scenarios, similar patterns are observed. In these cases, the graphs display a sinusoidal pattern with a diminishing amplitude with time. Surprisingly, in the phase space of the damped pendulum, a spiral type of graph is observed, demonstrating the intricate relationship between damping effects and nonlinearity. This research emphasizes the separatrix’s function as a crucial cutoff point where the motion of the pendulum changes from linear to chaotic.