Two-Coupled Pendulum System: Bifurcation, Chaos and the Potential Landscape Approach

In this paper, we have explored the bifurcation behavior and chaos of a two-coupled pendulum system with a coupling energy of the form: κ(θ 1 - θ 2 ) 2 . Using fixed point analysis, we have determined the bifurcation map, which provides a pictorial view of the number and stability properties of the fixed points with respect to the coupling parameter. The bifurcation map shows that the two-coupled pendulum system can exhibit three forms of bifurcation: pitchfork; saddle-node; and a new bifurcation in which a fixed point of the mixed type changes to a center, while a second fixed point of mixed type is born. In order to analyze the chaotic dynamics of the two-coupled pendulum system, we have introduced an equivalent model to the system. This model enables us to investigate the system dynamics in terms of the motion of a particle interacting with a potential landscape. Through analyzing the geometry of the landscape, we are able to determine the dynamical transition points from regular to locally chaotic, and then to globally chaotic behavior. The validity of our analytical prediction of E 0 = 4ε for the onset of chaos, and E T = π 2 ε/2 for the global transition to chaos, has been duly verified through numerical simulations.