Numerical simulation and geometrical analysis on the onset of chaos in a system of two coupled pendulums

By means of two geometrical approaches: (i) the potential landscape, and (ii) the Hamiltonian manifold method, we have found the relations that predict the transition to chaos in a two-coupled pendulum system due to a shift from positive to negative curvature. Our scrutiny from numerical simulations show that the splitting of the KAM island at the symmetric mode is another scenario that hints at the incidence of chaotic behaviour. We have uncovered two regimes on the onset of chaos, depending on whether the coupling parameter is greater or less than a critical point. In the first regime, chaos is observed to first occur after the splitting of the KAM island at the symmetric mode and before the switching of the dynamical curvature. On the other hand, an interchange of scenarios arises in the second regime, with chaos now appears after curvature switching and before KAM island splitting at the symmetric mode.