An approximate technique to test chaotic region in a rotating pendulum system with bistable characteristics

Abstract

This paper is concerned with the chaotic dynamics of a rotating pendulum system with bistable characteristics subjected to a viscous damping and a harmonic forcing. As a prototype of the single-degree-of-freedom system with bistable characteristics, this pendulum system exhibits a transition from smooth to discontinuous dynamics by changing a geometrical parameter. The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively. For the perturbed smooth system, we present an approximate technique to analytically obtain the lower bound line for horseshoes chaos arising from the homoclinic orbits of Duffing-type and Pendulum-type tangling, which overcomes the natural difficulties of solving the analytical expression of the homoclinic orbits and calculating the complicated Melnikov integrals. The chaotic thresholds of the perturbed discontinuous system are calculated by applying the numerical technique due to its non-smooth feature. Numerical simulations are carried out to certify the chaotic thresholds, which show the efficiency of the proposed techniques and demonstrate the predicated chaotic motions. Finally, different types of chaotic motions are illustrated via the cylindrical phase portraits. The contribution of this study is also helpful for exploring the dynamical behaviors of the complex nonlinear dynamical system containing the standard homoclinic or heteroclinic orbit in terms of the quantitative calculation.