Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation

Abstract

Experimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle θ to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed θ less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity mappings due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective damping value, a striking quantitative agreement is found too.