Possible chaotic pulsations in ZZ Ceti and rapidly oscillating Ap stars due to non-linear harmonic mode coupling

Summary Some ZZ Ceti and rapidly oscillating Ap stars show some multi-periodic behaviour, including prominent peaks in their power spectra at harmonically spaced frequencies with ratios 1 :2:3. The non-linear interaction between three such modes is a combination of the usual three-wave interaction coupled to second-harmonic generation. The amplitude equations for such a combined interaction are fundamentally different, and the Painleve analysis carried out here shows them to be non-integrable in the Hamiltonian sense, so that for long-term behaviour one expects irregular or chaotic phenomena instead of simple periodic or multi-periodic pulsations, especially when other modes and dissipation are included. Some examples of stars to which this might apply are given. These include VY Hor = BPM 3 1594 and BG CVn = GD 154 among the ZZ Ceti stars, PG 1351 + 489 among the DB variable white dwarfs, and HD 60435 and HD 22374 among the rapidly oscillating Ap stars.