The oscillating periodic solutions of a classical pendulum system with smooth and discontinuous dynamics

Abstract

The oscillating periodic solutions of a classical pendulum system with an irrational and fractional nonlinear restoring force are studied both theoretically and numerically under sufficiently small perturbations of a viscous damping and a harmonic excitation. The most salient feature of this pendulum system is to exhibit both smooth and discontinuous dynamics depending on the value of a geometrical parameter. In order to precisely describe the local dynamics of small-angle oscillations, we introduce a simplified approximate system which not only successfully retains the non-smooth characteristics but also completely reflects the local feature of the complex restoring force, especially the equilibrium bifurcation. Compared with the cubic and quintic polynomial systems derived by Taylor expansion, the application range of the simplified approximate system is enlarged within same margin of absolute error. With the help of the simplified approximate system, the periodic oscillatory solution around a stable equilibrium is examined analytically by using the averaging method in both smooth and discontinuous cases. Numerical simulations are carried out to verify the theoretical analysis and demonstrate the predicted periodic motions. The contribution of this study is to present an effective approximation to precisely describe the local dynamics of a classical pendulum system with smooth and discontinuous dynamics in terms of the qualitative analysis and quantitative calculation, which is also helpful for exploring the local dynamics of the nonlinear dynamical system containing a coupling of the irrational term and trigonometric function.