parametric resonance induced chaos in magnetic damped driven pendulum

A damped driven pendulum with a magnetic driving force, appearing from a solenoid, where ac current flows is considered. The solenoid acts on the magnet, which is located at the free end of the pendulum. In this system, the existence and interrelation of chaos and parametric resonance is theoretically examined. Derived analytical results are supported by numerical simulations and conducted experiments. In the present paper driving force is position angle dependent, particularly I consider a realistic example of driven damped pendulum model, via introducing magnetically driven case. In the context of magnetic pendulum, driving force is of a magnetic origin, particularly a solenoid with ac current is acting on the magnet, which plays a role of a bob in a pendulum with a rigid rod. Therefore the amplitude of a harmonic force greatly depends on the distance between solenoid and the magnet, making it angle dependent in a non-trivial form. In the frames of this model a possibility of onset of chaos has been examined analytically, numerically and experimentally. In our case (when orientation of solenoid’s dipolar moment is parallel to pendulum in unperturbed position) for some values of ac field and/or distance between solenoid and magnet chaos is observed due to the parametric resonance. Thus the main peculiarity of our model is that the existence of parametric resonance is a necessary condition for the onset of chaos in the system and in order to study chaotic behavior, Lyapunov exponents were calculated using numerical simulation and were compared to theoretical growth rate. Besides that, theory is compared to both numerical simulations and my own experiments.