Researchers studied how a simple symmetry in a driven oscillator leaves a detectable trace even when the motion becomes chaotic. The system is the driven Duffing–Holmes oscillator, a standard model that can sit in one well of a double-well potential or jump between both wells. The equation is unchanged if you flip the sign of the state and advance time by half a driving period. When the motion actually follows this rule, it is called antiperiodic: the state at time t is the point reflection (sign reversed) of the state at t plus half a drive period.

The team compared two ways of looking for repeats in a recorded trajectory. A recurrence plot (RP) marks times when the system returns near a previously visited state. That standard plot cannot see the sign-flip plus half-period symmetry because it only compares states to themselves. The authors therefore built an anti-recurrence plot (anti-RP). This chart compares the trajectory with its point-reflected copy: it flags times when the state at one time is close to the negative of the state at another time. In formulas, a point is marked when the distance between z_i and −z_j is below a chosen threshold. For a perfect antiperiodic solution the anti-RP shows diagonal lines offset by half a drive period.

They tested this on four regimes of the Duffing–Holmes oscillator at driving frequency ω = 1.3 and four driving amplitudes γ: a periodic single-well orbit (γ = 0.21), a chaotic single-well attractor (γ = 0.36), a periodic antiperiodic two-well orbit (γ = 0.637), and a chaotic two-well attractor (γ = 0.50). Time series were computed with a high-accuracy integrator from the same initial condition, discarding a long transient and sampling many points per drive period. Using the same geometric threshold for both matrices, standard recurrence metrics (for example the longest diagonal fraction Lmax/N and divergence DIV) separated order from chaos as usual. The anti-RP, however, cleanly distinguished whether the attractor occupied one well or visited both in the mirror-symmetric fashion.