Experimental Study of Bifurcations in A Parametrically Forced Pendulum

Abstract

An experimental study of bifurcations associated with stability of stationary points (SP's) in a parametrically forced magnetic pendulum and a comparison of its results with numerical results are presented. The critical values for which the SP's lose or gain their stability are experimentally measured by varying the two parameters $Ω$ (the normalized natural frequency) and $A$ (the normalized driving amplitude). It is observed that, when the amplitude $A$ exceeds a critical value, the normal SP with $θ=0$ ($θ$ is the angle between the permanent magnet and the magnetic field) becomes unstable either by a period-doubling bifurcation or by a symmetry-breaking pitchfork bifurcation, depending on the values of $Ω$. However, in contrast with the normal SP the inverted SP with $θ=π$ is observed to become stable as $A$ is increased above a critical value by a pitchfork bifurcation, but it also destabilizes for a higher critical value of $A$ by a period-doubling bifurcation. All of these experimental results agree well with numerical results obtained using the Floquet theory.