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Signatures of nonlinear mode interactions in the pulsating hot B subdwarf star KIC 10139564

We analyse 38-month of contiguous short-cadence data, concentrating on mode multiplets induced by the star rotation and on frequencies forming linear combinations that show intriguing behaviors during the course of the observations. We find clear signatures that point toward nonlinear effects predicted by resonant mode coupling mechanisms. These couplings can induce various mode behaviors for the components of multiplets and for frequencies related by linear relationships. We find that a triplet at 5760\,$μ$Hz, a quintuplet at 5287\,$μ$Hz and a ($\ell>2$) multiplet at 5412\,$μ$Hz, all induced by rotation, show clear frequency and amplitude modulations which are typical of the so-called intermediate regime of a resonance between the components. One triplet at 316\,$μ$Hz and a doublet at 394\,$μ$Hz show modulated amplitude and constant frequency which can be associated with a narrow transitory regime of the resonance. Another triplet at 519\,$μ$Hz appears to be in a frequency lock regime where both frequency and amplitude are constant. Additionally, three linear combination of frequencies near 6076\,$μ$Hz also show amplitude and frequency modulations, which are likely related to a three-mode direct resonance. The identified frequency and amplitude modulations are the first clear-cut signatures of nonlinear resonant couplings occurring in pulsating hot B subdwarf stars. However, the observed behaviors suggest that the resonances occurring in these stars usually follow more complicated patterns than the simple predictions from current nonlinear theoretical frameworks. These results should therefore motivate further work to develop the theory of nonlinear stellar pulsations, considering that stars like KIC~10139564 now offer remarkable testbeds to do so.