Exploring the unexpected: Disharmonized harmonic pulsation modes in high-amplitude $\delta$ Scuti stars

Harmonics are a ubiquitous feature across various pulsating stars. They are traditionally viewed as mere replicas of the independent primary pulsation modes and have thus been excluded from asteroseismological models. Recent research, however, has uncovered a significant discrepancy: in high-amplitude $\delta$ Scuti (HADS) stars, harmonics exhibit uncorrelated variations in amplitude and frequency relative to their independent primary pulsation modes. The nature of these disharmonized harmonics is a question of critical importance. In our study we analysed five triple-mode HADS stars observed by the Transiting Exoplanet Survey Satellite (TESS) and discovered some pervasive patterns of disharmonized harmonics in both the fundamental ($f_0$) and first overtone ($f_1$) pulsation modes. Intriguingly, through an in-depth frequency interaction analysis of V1393 Cen, we identified $2f_1$ as an independent pulsation mode, distinct from $f_1$, and identified it as the progenitor of the frequency variations observed in $3f_1$, $4f_1$, $5f_1$, and $6f_1$. Similar behaviour can be found in DO CMi and GSC 06047-00749, in which $2f_1$ and $3f_1$ are the independent pulsation modes, respectively. Notably, we found an interesting pattern when decomposing the harmonics that might suggest a generation process of harmonics. These findings serve as a new window on the research of harmonics, which remains a hidden corner of contemporary asteroseismology.