Global bifurcations of a rotating pendulum with irrational nonlinearity

• Display multiple dynamic bifurcations and their transitions as the parameters vary.
• Give the number, position and stability of the oscillating and rotational limit cycles.
• Find five limit cycles due to two saddle–node bifurcation of periodic orbits existing.
• Understand the formulation process of saddle–node bifurcation and rotational limit cycles.
• Unlike smooth case, the discontinuous case has no saddle–node bifurcation of periodic orbits.

In this paper, the authors consider a rotating pendulum under nonlinear perturbation which allows us to study various kinds of bifurcations and limit cycles. This system exhibits smooth or discontinuous dynamics depending on the value of a mechanical parameter. It is shown that the perturbed smooth system undergoes a pitchfork bifurcation, a homo–heteroclinic orbits transition, a single Hopf bifurcation, a double Hopf bifurcation, a pair of homoclinic bifurcations, a Hopf–homoclinic bifurcation and two saddle–node bifurcation of periodic orbits. The number, position and stability of all the oscillating and rotational limit cycles are given as the parameters vary. We find five limit cycles in the smooth case due to two saddle–node bifurcation of periodic orbits existing. Unlike the smooth case, the perturbed discontinuous system has a homoclinic-like bifurcation and without any saddle–node bifurcation of periodic orbits. Additionally, some results obtained herein bear significant similarities to the codimension-two bifurcation of duffing oscillator and SD oscillator, which may be helpful to explore the codimension bifurcation of the cylindrical pendulum system with irrational nonlinearity.