Rotating pendulum with smooth and discontinuous dynamics

Abstract

This paper focuses on a novel rotating pendulum system which is derived from the coupling rotational model of simple pendulum and mass-spring oscillator. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting smooth and discontinuous characteristics due to the geometry configuration. The equilibrium stability and bifurcations of the unperturbed system are explored showing the subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and saddle-node bifurcation. Furthermore, the complex transitions of multiple well dynamics and various kinds of singular orbits are demonstrated by using nonlinear dynamical technique in both smooth and discontinuous case. For the perturbed system, the primary resonance of small angle oscillation is investigated analytically by introducing the complete Jacobian elliptic integrals. Finally, complicated resonant structures of period, quasi-period and stochastic phenomena are presented for the system with unique harmonic perturbation. Bifurcation diagram, Poincaré section and basin analysis are carried out to demonstrate the co-existing different types of periodic motion and the transitions of chaotic motion between smooth and discontinuous case.