Recurrence and anti-recurrence patterns reveal an antiperiodic fingerprint that survives into chaos in the Duffing--Holmes oscillator

Abstract

The periodically forced Duffing--Holmes oscillator possesses a discrete symmetry under sign reversal of the coordinate combined with a half-period shift of the drive. When this symmetry is dynamically realized, the system supports \emph{antiperiodic} solutions, whose state at any instant is the point reflection of the state half a driving period earlier. We show that a standard recurrence plot (RP) is blind to this symmetry, whereas a complementary \emph{anti-recurrence plot} (anti-RP), built from the cross recurrence between a trajectory and its point-reflected image, detects it directly. Across four regimes -- periodic and chaotic single-well motion, and antiperiodic and chaotic two-well motion -- the anti-RP is empty when the attractor occupies one well and densely diagonal when the motion respects the symmetry. Crucially, the antiperiodic fingerprint persists into the chaotic two-well regime, where the anti-recurrence rate stays high relative to the ordinary one ($\mathrm{RR}_a/\mathrm{RR}\approx0.8$) despite the chaos. Recurrence quantification of both matrices separates order from chaos, while the anti-RP independently distinguishes one- from two-well, symmetry-respecting dynamics, giving a compact classification of all regimes. Requiring only a time series and the symmetry operation, the anti-RP is a model-free probe of dynamical symmetry for any system with a sign-reversal invariance, including experimental signals where phase-averaged observables fail.