Periodic Motions and Bifurcations of a Periodically Forced Spring Pendulum Varying With Excitation Amplitude

The spring pendulum system is one of the typical two degree of freedom (2-DOF) oscillating system with nonlinear coupling. Such system has long been of interest, and has been attracting researchers from all over the world. Such system is very simple to construct, yet it possesses very complicated dynamical behaviors. In this paper, the method of implicit discrete maps will be employed to investigate the nonlinear dynamical behavior of a spring pendulum with nonlinear stiffness. Previously, such method has been successfully applied to many continuous nonlinear dynamical systems, such as Duffing oscillation system, periodically excited pendulum, etc. Such method uses discrete mapping structure to represent different periodic motions in a dynamical system. And semi-analytical predictions of periodic motions can then be obtained numerically from the system of governing equations of the discrete maps. And thus, bifurcation trees of various periodic motions can be analytically predicted using computer program. Thus, in this paper, implicit discrete maps are obtained from the midpoint scheme of the corresponding differential equations of a nonlinear spring pendulum system. Discrete mapping structures are developed to represent different periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved to obtain predictions of the bifurcation trees. With varying the excitation amplitude, the complete bifurcations of period-1 motion to chaos is investigated for the spring pendulum system. Such bifurcation behaviors are demonstrated with the bifurcation tree of period-1 to period-2 motions. The corresponding stability are studied through eigenvalue analysis on the Jacobian matrix of the mapping structure. Bifurcation conditions are also obtained through such eigenvalues. Various types of bifurcations are observed. The cascaded period doubling bifurcations lead to period-2, period-4, period-8, and period-16… motions. High period motions are not demonstrated to avoid redundancy due to the abundant data. Neimark bifurcations are observed with transition from stable to unstable. Furthermore, three types of saddle node bifurcations are observed: Saddle node bifurcations associated with asymmetric motions, saddle node bifurcation associated to jumping phenomenon with stable to unstable, and saddle node bifurcations associated to jumping phenomenon with unstable to unstable. The corresponding bifurcation trees of harmonic amplitudes are also presented to shown the motion complexity and nonlinearity in such a nonlinear spring pendulum system. Such harmonic amplitudes also demonstrate the bifurcation behaviors of the system under frequency domain. Finally, numerical illustrations of various periodic motions are obtained through both simulations and analytical predictions, for which the parameters are picked from the bifurcation trees presented. Such periodic motions are illustrated for comparison between the simulated and predicted orbits as a verification of the analytical predictions.