Chaotic orbital dynamics of pulsating stars around black holes surrounded by dark matter halos

We analyze the orbital dynamics of spherical test bodies in ``black hole surrounded by dark matter halo'' spherically symmetric spacetimes. When the test body pulsates periodically (such as a variable star), altering its quadrupole tensor, Melnikov's method shows that its orbital dynamics presents homoclinic chaos near the corresponding unstable circular orbits however small the oscillation amplitude is. Since for supermassive black holes the period of revolution of a star near the innermost stable circular orbit roughly spans time intervals from minutes to hours, the formalism can be applied in principle to the astrophysical scenario of a pulsating (variable) star inspiraling into a supermassive black hole, including the black hole SgrA* at the center of our Galaxy. The chaotic nature of its orbit, due to pulsation, is imprinted in the redshift time series of the emitted light and can, in principle, be observed in the corresponding light curves and even in gravitational-wave signals detected by future observatories such as the Laser Inteferometer Space Antenna. Also, although periodic with respect to the star's proper time, the chaotic orbital motion will produce an erratic light curve (and gravitational-wave signal) in terms of observed, coordinate time. Although our results were obtained for a specific exact solution, we argue that this phenomenon is generic for pulsating bodies immersed in black hole spacetimes surrounded by self-gravitating fluids.