Complex dynamics and chaos in commutable pendulum

This paper deals with a commutable pendulum which has two different natural frequencies in two space regions. The first region corresponds to the first three quadrants of the phase space; while the second region coincides with the fourth quadrant. This apparently trivial system shows a very complex behavior, regardless of the fact that it is based on the simple pendulum model. In the case where the two natural frequencies coincide, the flow is stable. However, just by varying the natural frequency in the fourth quadrant, the system may be asymptotically stable or unstable and also may have simple limit cycle, complex limit cycles and even chaotic behavior. In all these cases the trajectories have simple analytical descriptions.