Antiperiodicity in the Duffing--Holmes oscillator: symmetry origin, parity selection, and transitions
We investigate the origin and distribution of antiperiodicity --- oscillations satisfying $x(t+T)=-x(t)$ --- in the periodically driven Duffing--Holmes oscillator, combining analytical arguments with extensive numerical exploration. Antiperiodic orbits are precisely the periodic orbits invariant under the half-period shift symmetry $S:(x,\dot{x},t)\mapsto(-x,-\dot{x},\,t+T_d/2)$ of the equations of motion, with $T_d$ the driving period. We map the antiperiodic regions across the plane spanned by the amplitude and the frequency of the forcing, together with the periodic and chaotic domains and the potential wells visited by each orbit. The invariance under $S$ imposes a parity selection rule, verified without exception across our parameter sweeps: antiperiodic orbits lock to the drive only at odd multiples of the forcing period. Periodic orbits that lack the antisymmetry occur instead as conjugate pairs related by $S$, each orbit being the point reflection of its twin. We further show that antiperiodic orbits cannot bifurcate through a direct period doubling: the symmetry must break first, in a supercritical pitchfork in which the antiperiodic orbit splits into two conjugate, symmetry-broken orbits; alternatively, the antiperiodic orbit disappears with its symmetry intact, in a saddle-node bifurcation with an antiperiodic saddle. Antiperiodicity thus emerges as the orbit-level manifestation of a discrete symmetry of the driven system.