Investigation of chaos behaviour on damped and driven nonlinear simple pendulum motion simulated by mathematica

Abstract

This study aimed to investigate the chaos behavior resulted from damped driven nonlinear simple pendulum motion, which was simulated by using Mathematica. The method used to solve the equation of the pendulum was the 4th order of Runge-Kutta. The solution was obtained by setting some parameters to be determined at a certain value (dimensionless), namely the gravitational acceleration (g) was 9.8; the length of rope (l) was 9.8, the dumpling coefficient (q) was 0.4; the initial conditions (θ 0 and ω 0 ) were 0.8 for both; the frequency of driving force (Ω D ) was 0.6; and the mass of pendulum (m) was 1. Due to these settings, the natural frequency of the pendulum (Ω 2 ) was calculated to be 1. Still, the driving force (a) was varied in order to analyze the chaos clearly. The solution was then plotted as θ vs t graphs, phase space, and Poincaré section. The results showed that the chaotic motion occurs when the driving force was in the range of 1.36-3.0, in which 3.0 was the maximum value in this simulation. In this range, the slight change in the initial condition resulted in significant differences of θ vs t graph, implying sensitivity to initial conditions. The phase space depicted a chaotic attractor, while the Poincaré section resulted in many dots forming stretching and folding patterns. Based on these results, it can be said that the chaos behavior could arise from damped and driven nonlinear simple pendulum motion by varying parameters, such as driving force.