Symmetry and Dynamics of A Magnetic Oscillator

We consider a permanent magnetic dipole in an oscillating magnetic field. This magnetic oscillator has two dynamical symmetries. With increasing the amplitude $A$ of the magnetic field, dynamical behaviors associated with the symmetries are investigated. For small $A$, there exist symmetric states with respect to one of the two symmetries. However, such symmetric states lose their symmetries via symmetry-breaking pitchfork bifurcations and then the symmetry-broken states exhibit period-doubling transitions to chaos. Consequently, small chaotic attractors with broken symmetries appear. However, as $A$ is further increased they merge into a large symmetric chaotic attractor via symmetry-restoring attractor-merging crisis.