The cradle of nonlinear asteroseismology: observations of oscillation mode variability in compact pulsating stars

We briefly review progress in developing a pathway to nonlinear astereoseismology, both from theoretical and observational aspects. As predicted by the theory of weak nonlinear interactions between resonant modes, their amplitude and frequency can be modulated according to various kinds of patterns. However, those subtle modulations could hardly be well characterized from ground-based photometric monitoring. The {\sl Kepler} spacecraft offered a new window to find clear-cut evidence of well-determined amplitude and frequency modulations, leading to the first discoveries of such variations in pulsating white dwarf and hot B subdwarf stars. Following that direction, a systematic survey of oscillation mode properties in compact pulsators monitored by {\sl Kepler} suggests that mode variability is likely a common phenomenon, which remain unaccounted for by standard linear non-radial pulsation theory. To reach this conclusion firmly, the survey has now been extended to a larger context including compact stars observed by K2 and TESS. We expect that this extended survey will help to constrain key parameters governing weak nonlinear effects in stellar oscillations.