Resonance response and chaotic analysis for an irrational pendulum system

We propose a novel rotating pendulum system with irrational nonlinearity, bi-stable phenomenon and quasi-zero stiffness characteristic, which bears a transition from standard double well dynamics to discontinuous double well dynamics by adjusting a geometric parameter. Nonlinear dynamical behaviors of the proposed irrational pendulum system subjected to viscus damping and periodic excitation are studied. To analytically demonstrate the primary resonance response for this perturbed irrational pendulum system, we introduce an approximate irrational system which is significant parallels with the original pendulum system from the perspective of qualitative analysis and quantitative calculation. Averaging method is applied to investigate the dynamic response of the perturbed irrational pendulum system with quasi-zero stiffness at origin. The effect of the internal and external parameters on the response curves are discussed. Numerical fitting technique and semi-analytical Melnikov method are used to detect the chaotic boundaries of the perturbed irrational pendulum system with two types of homoclinic orbits. Different types of periodic solutions and chaos are clarified in the perturbed irrational pendulum system by using numerical simulations. It is found that a class of chaos with oscillatory and rotational motions bears significant similarities to paroxysmal chaos due to sudden rotational motion and two chaotic motions formed by period-doubling bifurcation merge into one chaos.