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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(θ1 - θ2)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 the...